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Sparsity-certifying Graph Decompositions Ileana Streinu1∗, Louis Theran2 1 Department of Computer Science, Smith College, Northampton, MA. e-mail: streinu@cs.smith.edu 2 Department of Computer Science, University of Massachusetts Amherst. e-mail: theran@cs.umass.edu Abstract. We describe a new algorithm, the (k, `)-pebble game with colors, and use it to obtain a charac- terization of the family of (k, `)-sparse graphs and algorithmic solutions to a family of problems concern- ing tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characteri- zation of arboricity. We also present a new decomposition that certifies sparsity based on the (k, `)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9]. 1. Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems

We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use it obtain a characterization of the family of $(k,\ell)$-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the $(k,\ell)$-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and Westermann and Hendrickson.

Introduction and preliminaries The focus of this paper is decompositions of (k, `)-sparse graphs into edge-disjoint subgraphs that certify sparsity. We use graph to mean a multigraph, possibly with loops. We say that a graph is (k, `)-sparse if no subset of n′ vertices spans more than kn′− ` edges in the graph; a (k, `)-sparse graph with kn′− ` edges is (k, `)-tight. We call the range k ≤ `≤ 2k−1 the upper range of sparse graphs and 0≤ `≤ k the lower range. In this paper, we present efficient algorithms for finding decompositions that certify sparsity in the upper range of `. Our algorithms also apply in the lower range, which was already ad- dressed by [3, 4, 5, 6, 19]. A decomposition certifies the sparsity of a graph if the sparse graphs and graphs admitting the decomposition coincide. Our algorithms are based on a new characterization of sparse graphs, which we call the pebble game with colors. The pebble game with colors is a simple graph construction rule that produces a sparse graph along with a sparsity-certifying decomposition. We define and study a canonical class of pebble game constructions, which correspond to previously studied decompositions of sparse graphs into edge disjoint trees. Our results provide a unifying framework for all the previously known special cases, including Nash-Williams- Tutte and [7, 24]. Indeed, in the lower range, canonical pebble game constructions capture the properties of the augmenting paths used in matroid union and intersection algorithms[5, 6]. Since the sparse graphs in the upper range are not known to be unions or intersections of the matroids for which there are efficient augmenting path algorithms, these do not easily apply in ∗ Research of both authors funded by the NSF under grants NSF CCF-0430990 and NSF-DARPA CARGO CCR-0310661 to the first author. 2 Ileana Streinu, Louis Theran Term Meaning Sparse graph G Every non-empty subgraph on n′ vertices has ≤ kn′− ` edges Tight graph G G = (V,E) is sparse and |V |= n, |E|= kn− ` Block H in G G is sparse, and H is a tight subgraph Component H of G G is sparse and H is a maximal block Map-graph Graph that admits an out-degree-exactly-one orientation (k, `)-maps-and-trees Edge-disjoint union of ` trees and (k− `) map-grpahs `Tk Union of ` trees, each vertex is in exactly k of them Set of tree-pieces of an `Tk induced on V ′ ⊂V Pieces of trees in the `Tk spanned by E(V ′) Proper `Tk Every V ′ ⊂V contains ≥ ` pieces of trees from the `Tk Table 1. Sparse graph and decomposition terminology used in this paper. the upper range. Pebble game with colors constructions may thus be considered a strengthening of augmenting paths to the upper range of matroidal sparse graphs. 1.1. Sparse graphs A graph is (k, `)-sparse if for any non-empty subgraph with m′ edges and n′ vertices, m′ ≤ kn′− `. We observe that this condition implies that 0 ≤ ` ≤ 2k− 1, and from now on in this paper we will make this assumption. A sparse graph that has n vertices and exactly kn−` edges is called tight. For a graph G = (V,E), and V ′ ⊂ V , we use the notation span(V ′) for the number of edges in the subgraph induced by V ′. In a directed graph, out(V ′) is the number of edges with the tail in V ′ and the head in V −V ′; for a subgraph induced by V ′, we call such an edge an out-edge. There are two important types of subgraphs of sparse graphs. A block is a tight subgraph of a sparse graph. A component is a maximal block. Table 1 summarizes the sparse graph terminology used in this paper. 1.2. Sparsity-certifying decompositions A k-arborescence is a graph that admits a decomposition into k edge-disjoint spanning trees. Figure 1(a) shows an example of a 3-arborescence. The k-arborescent graphs are described by the well-known theorems of Tutte [23] and Nash-Williams [17] as exactly the (k,k)-tight graphs. A map-graph is a graph that admits an orientation such that the out-degree of each vertex is exactly one. A k-map-graph is a graph that admits a decomposition into k edge-disjoint map- graphs. Figure 1(b) shows an example of a 2-map-graphs; the edges are oriented in one possible configuration certifying that each color forms a map-graph. Map-graphs may be equivalently defined (see, e.g., [18]) as having exactly one cycle per connected component.1 A (k, `)-maps-and-trees is a graph that admits a decomposition into k− ` edge-disjoint map-graphs and ` spanning trees. Another characterization of map-graphs, which we will use extensively in this paper, is as the (1,0)-tight graphs [8, 24]. The k-map-graphs are evidently (k,0)-tight, and [8, 24] show that the converse holds as well. 1 Our terminology follows Lovász in [16]. In the matroid literature map-graphs are sometimes known as bases of the bicycle matroid or spanning pseudoforests. Sparsity-certifying Graph Decompositions 3 Fig. 1. Examples of sparsity-certifying decompositions: (a) a 3-arborescence; (b) a 2-map-graph; (c) a (2,1)-maps-and-trees. Edges with the same line style belong to the same subgraph. The 2-map-graph is shown with a certifying orientation. A `Tk is a decomposition into ` edge-disjoint (not necessarily spanning) trees such that each vertex is in exactly k of them. Figure 2(a) shows an example of a 3T2. Given a subgraph G′ of a `Tk graph G, the set of tree-pieces in G′ is the collection of the components of the trees in G induced by G′ (since G′ is a subgraph each tree may contribute multiple pieces to the set of tree-pieces in G′). We observe that these tree-pieces may come from the same tree or be single-vertex “empty trees.” It is also helpful to note that the definition of a tree-piece is relative to a specific subgraph. An `Tk decomposition is proper if the set of tree-pieces in any subgraph G′ has size at least `. Figure 2(a) shows a graph with a 3T2 decomposition; we note that one of the trees is an isolated vertex in the bottom-right corner. The subgraph in Figure 2(b) has three black tree- pieces and one gray tree-piece: an isolated vertex at the top-right corner, and two single edges. These count as three tree-pieces, even though they come from the same back tree when the whole graph in considered. Figure 2(c) shows another subgraph; in this case there are three gray tree-pieces and one black one. Table 1 contains the decomposition terminology used in this paper. The decomposition problem. We define the decomposition problem for sparse graphs as tak- ing a graph as its input and producing as output, a decomposition that can be used to certify spar- sity. In this paper, we will study three kinds of outputs: maps-and-trees; proper `Tk decompositions; and the pebble-game-with-colors decomposition, which is defined in the next section. 2. Historical background The well-known theorems of Tutte [23] and Nash-Williams [17] relate the (k,k)-tight graphs to the existence of decompositions into edge-disjoint spanning trees. Taking a matroidal viewpoint, 4 Ileana Streinu, Louis Theran Fig. 2. (a) A graph with a 3T2 decomposition; one of the three trees is a single vertex in the bottom right corner. (b) The highlighted subgraph inside the dashed countour has three black tree-pieces and one gray tree-piece. (c) The highlighted subgraph inside the dashed countour has three gray tree-pieces (one is a single vertex) and one black tree-piece. Edmonds [3, 4] gave another proof of this result using matroid unions. The equivalence of maps- and-trees graphs and tight graphs in the lower range is shown using matroid unions in [24], and matroid augmenting paths are the basis of the algorithms for the lower range of [5, 6, 19]. In rigidity theory a foundational theorem of Laman [11] shows that (2,3)-tight (Laman) graphs correspond to generically minimally rigid bar-and-joint frameworks in the plane. Tay [21] proved an analogous result for body-bar frameworks in any dimension using (k,k)-tight graphs. Rigidity by counts motivated interest in the upper range, and Crapo [2] proved the equivalence of Laman graphs and proper 3T2 graphs. Tay [22] used this condition to give a direct proof of Laman’s theorem and generalized the 3T2 condition to all `Tk for k≤ `≤ 2k−1. Haas [7] studied `Tk decompositions in detail and proved the equivalence of tight graphs and proper `Tk graphs for the general upper range. We observe that aside from our new pebble- game-with-colors decomposition, all the combinatorial characterizations of the upper range of sparse graphs, including the counts, have a geometric interpretation [11, 21, 22, 24]. A pebble game algorithm was first proposed in [10] as an elegant alternative to Hendrick- son’s Laman graph algorithms [9]. Berg and Jordan [1], provided the formal analysis of the pebble game of [10] and introduced the idea of playing the game on a directed graph. Lee and Streinu [12] generalized the pebble game to the entire range of parameters 0≤ `≤ 2k−1, and left as an open problem using the pebble game to find sparsity certifying decompositions. 3. The pebble game with colors Our pebble game with colors is a set of rules for constructing graphs indexed by nonnegative integers k and `. We will use the pebble game with colors as the basis of an efficient algorithm for the decomposition problem later in this paper. Since the phrase “with colors” is necessary only for comparison to [12], we will omit it in the rest of the paper when the context is clear. Sparsity-certifying Graph Decompositions 5 We now present the pebble game with colors. The game is played by a single player on a fixed finite set of vertices. The player makes a finite sequence of moves; a move consists in the addition and/or orientation of an edge. At any moment of time, the state of the game is captured by a directed graph H, with colored pebbles on vertices and edges. The edges of H are colored by the pebbles on them. While playing the pebble game all edges are directed, and we use the notation vw to indicate a directed edge from v to w. We describe the pebble game with colors in terms of its initial configuration and the allowed moves. Fig. 3. Examples of pebble game with colors moves: (a) add-edge. (b) pebble-slide. Pebbles on vertices are shown as black or gray dots. Edges are colored with the color of the pebble on them. Initialization: In the beginning of the pebble game, H has n vertices and no edges. We start by placing k pebbles on each vertex of H, one of each color ci, for i = 1,2, . . . ,k. Add-edge-with-colors: Let v and w be vertices with at least `+1 pebbles on them. Assume (w.l.o.g.) that v has at least one pebble on it. Pick up a pebble from v, add the oriented edge vw to E(H) and put the pebble picked up from v on the new edge. Figure 3(a) shows examples of the add-edge move. Pebble-slide: Let w be a vertex with a pebble p on it, and let vw be an edge in H. Replace vw with wv in E(H); put the pebble that was on vw on v; and put p on wv. Note that the color of an edge can change with a pebble-slide move. Figure 3(b) shows examples. The convention in these figures, and throughout this paper, is that pebbles on vertices are represented as colored dots, and that edges are shown in the color of the pebble on them. From the definition of the pebble-slide move, it is easy to see that a particular pebble is always either on the vertex where it started or on an edge that has this vertex as the tail. However, when making a sequence of pebble-slide moves that reverse the orientation of a path in H, it is sometimes convenient to think of this path reversal sequence as bringing a pebble from the end of the path to the beginning. The output of playing the pebble game is its complete configuration. Output: At the end of the game, we obtain the directed graph H, along with the location and colors of the pebbles. Observe that since each edge has exactly one pebble on it, the pebble game configuration colors the edges. We say that the underlying undirected graph G of H is constructed by the (k, `)-pebble game or that H is a pebble-game graph. Since each edge of H has exactly one pebble on it, the pebble game’s configuration partitions the edges of H, and thus G, into k different colors. We call this decomposition of H a pebble- game-with-colors decomposition. Figure 4(a) shows an example of a (2,2)-tight graph with a pebble-game decomposition. Let G = (V,E) be pebble-game graph with the coloring induced by the pebbles on the edges, and let G′ be a subgraph of G. Then the coloring of G induces a set of monochromatic con- 6 Ileana Streinu, Louis Theran (a) (b) (c) Fig. 4. A (2,2)-tight graph with one possible pebble-game decomposition. The edges are oriented to show (1,0)-sparsity for each color. (a) The graph K4 with a pebble-game decomposition. There is an empty black tree at the center vertex and a gray spanning tree. (b) The highlighted subgraph has two black trees and a gray tree; the black edges are part of a larger cycle but contribute a tree to the subgraph. (c) The highlighted subgraph (with a light gray background) has three empty gray trees; the black edges contain a cycle and do not contribute a piece of tree to the subgraph. Notation Meaning span(V ′) Number of edges spanned in H by V ′ ⊂V ; i.e. |EH(V ′)| peb(V ′) Number of pebbles on V ′ ⊂V out(V ′) Number of edges vw in H with v ∈V ′ and w ∈V −V ′ pebi(v) Number of pebbles of color ci on v ∈V outi(v) Number of edges vw colored ci for v ∈V Table 2. Pebble game notation used in this paper. nected subgraphs of G′ (there may be more than one of the same color). Such a monochromatic subgraph is called a map-graph-piece of G′ if it contains a cycle (in G′) and a tree-piece of G′ otherwise. The set of tree-pieces of G′ is the collection of tree-pieces induced by G′. As with the corresponding definition for `Tk s, the set of tree-pieces is defined relative to a specific sub- graph; in particular a tree-piece may be part of a larger cycle that includes edges not spanned by G′. The properties of pebble-game decompositions are studied in Section 6, and Theorem 2 shows that each color must be (1,0)-sparse. The orientation of the edges in Figure 4(a) shows this. For example Figure 4(a) shows a (2,2)-tight graph with one possible pebble-game decom- position. The whole graph contains a gray tree-piece and a black tree-piece that is an isolated vertex. The subgraph in Figure 4(b) has a black tree and a gray tree, with the edges of the black tree coming from a cycle in the larger graph. In Figure 4(c), however, the black cycle does not contribute a tree-piece. All three tree-pieces in this subgraph are single-vertex gray trees. In the following discussion, we use the notation peb(v) for the number of pebbles on v and pebi(v) to indicate the number of pebbles of colors i on v. Table 2 lists the pebble game notation used in this paper. 4. Our Results We describe our results in this section. The rest of the paper provides the proofs. Sparsity-certifying Graph Decompositions 7 Our first result is a strengthening of the pebble games of [12] to include colors. It says that sparse graphs are exactly pebble game graphs. Recall that from now on, all pebble games discussed in this paper are our pebble game with colors unless noted explicitly. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. Next we consider pebble-game decompositions, showing that they are a generalization of proper `Tk decompositions that extend to the entire matroidal range of sparse graphs. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. The (1,0)-sparse subgraphs in the statement of Theorem 2 are the colors of the pebbles; thus Theorem 2 gives a characterization of the pebble-game-with-colors decompositions obtained by playing the pebble game defined in the previous section. Notice the similarity between the requirement that the set of tree-pieces have size at least ` in Theorem 2 and the definition of a proper `Tk . Our next results show that for any pebble-game graph, we can specialize its pebble game construction to generate a decomposition that is a maps-and-trees or proper `Tk . We call these specialized pebble game constructions canonical, and using canonical pebble game construc- tions, we obtain new direct proofs of existing arboricity results. We observe Theorem 2 that maps-and-trees are special cases of the pebble-game decompo- sition: both spanning trees and spanning map-graphs are (1,0)-sparse, and each of the spanning trees contributes at least one piece of tree to every subgraph. The case of proper `Tk graphs is more subtle; if each color in a pebble-game decomposition is a forest, then we have found a proper `Tk , but this class is a subset of all possible proper `Tk decompositions of a tight graph. We show that this class of proper `Tk decompositions is sufficient to certify sparsity. We now state the main theorem for the upper and lower range. Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Theorem 4 (Main Theorem (Upper Range): Proper `Tk graphs coincide with pebble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. As corollaries, we obtain the existing decomposition results for sparse graphs. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . Efficiently finding canonical pebble game constructions. The proofs of Theorem 3 and Theo- rem 4 lead to an obvious algorithm with O(n3) running time for the decomposition problem. Our last result improves on this, showing that a canonical pebble game construction, and thus 8 Ileana Streinu, Louis Theran a maps-and-trees or proper `Tk decomposition can be found using a pebble game algorithm in O(n2) time and space. These time and space bounds mean that our algorithm can be combined with those of [12] without any change in complexity. 5. Pebble game graphs In this section we prove Theorem 1, a strengthening of results from [12] to the pebble game with colors. Since many of the relevant properties of the pebble game with colors carry over directly from the pebble games of [12], we refer the reader there for the proofs. We begin by establishing some invariants that hold during the execution of the pebble game. Lemma 7 (Pebble game invariants). During the execution of the pebble game, the following invariants are maintained in H: (I1) There are at least ` pebbles on V . [12] (I2) For each vertex v, span(v)+out(v)+peb(v) = k. [12] (I3) For each V ′ ⊂V , span(V ′)+out(V ′)+peb(V ′) = kn′. [12] (I4) For every vertex v ∈V , outi(v)+pebi(v) = 1. (I5) Every maximal path consisting only of edges with color ci ends in either the first vertex with a pebble of color ci or a cycle. Proof. (I1), (I2), and (I3) come directly from [12]. (I4) This invariant clearly holds at the initialization phase of the pebble game with colors. That add-edge and pebble-slide moves preserve (I4) is clear from inspection. (I5) By (I4), a monochromatic path of edges is forced to end only at a vertex with a pebble of the same color on it. If there is no pebble of that color reachable, then the path must eventually visit some vertex twice. From these invariants, we can show that the pebble game constructible graphs are sparse. Lemma 8 (Pebble-game graphs are sparse [12]). Let H be a graph constructed with the pebble game. Then H is sparse. If there are exactly ` pebbles on V (H), then H is tight. The main step in proving that every sparse graph is a pebble-game graph is the following. Recall that by bringing a pebble to v we mean reorienting H with pebble-slide moves to reduce the out degree of v by one. Lemma 9 (The `+1 pebble condition [12]). Let vw be an edge such that H + vw is sparse. If peb({v,w}) < `+1, then a pebble not on {v,w} can be brought to either v or w. It follows that any sparse graph has a pebble game construction. Theorem 1 (Sparse graphs and pebble-game graphs coincide). A graph G is (k, `)-sparse with 0≤ `≤ 2k−1 if and only if G is a pebble-game graph. 6. The pebble-game-with-colors decomposition In this section we prove Theorem 2, which characterizes all pebble-game decompositions. We start with the following lemmas about the structure of monochromatic connected components in H, the directed graph maintained during the pebble game. Sparsity-certifying Graph Decompositions 9 Lemma 10 (Monochromatic pebble game subgraphs are (1,0)-sparse). Let Hi be the sub- graph of H induced by edges with pebbles of color ci on them. Then Hi is (1,0)-sparse, for i = 1, . . . ,k. Proof. By (I4) Hi is a set of edges with out degree at most one for every vertex. Lemma 11 (Tree-pieces in a pebble-game graph). Every subgraph of the directed graph H in a pebble game construction contains at least ` monochromatic tree-pieces, and each of these is rooted at either a vertex with a pebble on it or a vertex that is the tail of an out-edge. Recall that an out-edge from a subgraph H ′ = (V ′,E ′) is an edge vw with v∈V ′ and vw /∈ E ′. Proof. Let H ′ = (V ′,E ′) be a non-empty subgraph of H, and assume without loss of generality that H ′ is induced by V ′. By (I3), out(V ′)+ peb(V ′) ≥ `. We will show that each pebble and out-edge tail is the root of a tree-piece. Consider a vertex v ∈ V ′ and a color ci. By (I4) there is a unique monochromatic directed path of color ci starting at v. By (I5), if this path ends at a pebble, it does not have a cycle. Similarly, if this path reaches a vertex that is the tail of an out-edge also in color ci (i.e., if the monochromatic path from v leaves V ′), then the path cannot have a cycle in H ′. Since this argument works for any vertex in any color, for each color there is a partitioning of the vertices into those that can reach each pebble, out-edge tail, or cycle. It follows that each pebble and out-edge tail is the root of a monochromatic tree, as desired. Applied to the whole graph Lemma 11 gives us the following. Lemma 12 (Pebbles are the roots of trees). In any pebble game configuration, each pebble of color ci is the root of a (possibly empty) monochromatic tree-piece of color ci. Remark: Haas showed in [7] that in a `Tk , a subgraph induced by n′ ≥ 2 vertices with m′ edges has exactly kn′−m′ tree-pieces in it. Lemma 11 strengthens Haas’ result by extending it to the lower range and giving a construction that finds the tree-pieces, showing the connection between the `+1 pebble condition and the hereditary condition on proper `Tk . We conclude our investigation of arbitrary pebble game constructions with a description of the decomposition induced by the pebble game with colors. Theorem 2 (The pebble-game-with-colors decomposition). A graph G is a pebble-game graph if and only if it admits a decomposition into k edge-disjoint subgraphs such that each is (1,0)-sparse and every subgraph of G contains at least ` tree-pieces of the (1,0)-sparse graphs in the decomposition. Proof. Let G be a pebble-game graph. The existence of the k edge-disjoint (1,0)-sparse sub- graphs was shown in Lemma 10, and Lemma 11 proves the condition on subgraphs. For the other direction, we observe that a color ci with ti tree-pieces in a given subgraph can span at most n− ti edges; summing over all the colors shows that a graph with a pebble-game decomposition must be sparse. Apply Theorem 1 to complete the proof. Remark: We observe that a pebble-game decomposition for a Laman graph may be read out of the bipartite matching used in Hendrickson’s Laman graph extraction algorithm [9]. Indeed, pebble game orientations have a natural correspondence with the bipartite matchings used in 10 Ileana Streinu, Louis Theran Maps-and-trees are a special case of pebble-game decompositions for tight graphs: if there are no cycles in ` of the colors, then the trees rooted at the corresponding ` pebbles must be spanning, since they have n− 1 edges. Also, if each color forms a forest in an upper range pebble-game decomposition, then the tree-pieces condition ensures that the pebble-game de- composition is a proper `Tk . In the next section, we show that the pebble game can be specialized to correspond to maps- and-trees and proper `Tk decompositions. 7. Canonical Pebble Game Constructions In this section we prove the main theorems (Theorem 3 and Theorem 4), continuing the inves- tigation of decompositions induced by pebble game constructions by studying the case where a minimum number of monochromatic cycles are created. The main idea, captured in Lemma 15 and illustrated in Figure 6, is to avoid creating cycles while collecting pebbles. We show that this is always possible, implying that monochromatic map-graphs are created only when we add more than k(n′−1) edges to some set of n′ vertices. For the lower range, this implies that every color is a forest. Every decomposition characterization of tight graphs discussed above follows immediately from the main theorem, giving new proofs of the previous results in a unified framework. In the proof, we will use two specializations of the pebble game moves. The first is a modi- fication of the add-edge move. Canonical add-edge: When performing an add-edge move, cover the new edge with a color that is on both vertices if possible. If not, then take the highest numbered color present. The second is a restriction on which pebble-slide moves we allow. Canonical pebble-slide: A pebble-slide move is allowed only when it does not create a monochromatic cycle. We call a pebble game construction that uses only these moves canonical. In this section we will show that every pebble-game graph has a canonical pebble game construction (Lemma 14 and Lemma 15) and that canonical pebble game constructions correspond to proper `Tk and maps-and-trees decompositions (Theorem 3 and Theorem 4). We begin with a technical lemma that motivates the definition of canonical pebble game constructions. It shows that the situations disallowed by the canonical moves are all the ways for cycles to form in the lowest ` colors. Lemma 13 (Monochromatic cycle creation). Let v ∈ V have a pebble p of color ci on it and let w be a vertex in the same tree of color ci as v. A monochromatic cycle colored ci is created in exactly one of the following ways: (M1) The edge vw is added with an add-edge move. (M2) The edge wv is reversed by a pebble-slide move and the pebble p is used to cover the reverse edge vw. Proof. Observe that the preconditions in the statement of the lemma are implied by Lemma 7. By Lemma 12 monochromatic cycles form when the last pebble of color ci is removed from a connected monochromatic subgraph. (M1) and (M2) are the only ways to do this in a pebble game construction, since the color of an edge only changes when it is inserted the first time or a new pebble is put on it by a pebble-slide move. Sparsity-certifying Graph Decompositions 11 vw vw Fig. 5. Creating monochromatic cycles in a (2,0)-pebble game. (a) A type (M1) move creates a cycle by adding a black edge. (b) A type (M2) move creates a cycle with a pebble-slide move. The vertices are labeled according to their role in the definition of the moves. Figure 5(a) and Figure 5(b) show examples of (M1) and (M2) map-graph creation moves, respectively, in a (2,0)-pebble game construction. We next show that if a graph has a pebble game construction, then it has a canonical peb- ble game construction. This is done in two steps, considering the cases (M1) and (M2) sepa- rately. The proof gives two constructions that implement the canonical add-edge and canonical pebble-slide moves. Lemma 14 (The canonical add-edge move). Let G be a graph with a pebble game construc- tion. Cycle creation steps of type (M1) can be eliminated in colors ci for 1 ≤ i ≤ `′, where `′ = min{k, `}. Proof. For add-edge moves, cover the edge with a color present on both v and w if possible. If this is not possible, then there are `+1 distinct colors present. Use the highest numbered color to cover the new edge. Remark: We note that in the upper range, there is always a repeated color, so no canonical add-edge moves create cycles in the upper range. The canonical pebble-slide move is defined by a global condition. To prove that we obtain the same class of graphs using only canonical pebble-slide moves, we need to extend Lemma 9 to only canonical moves. The main step is to show that if there is any sequence of moves that reorients a path from v to w, then there is a sequence of canonical moves that does the same thing. Lemma 15 (The canonical pebble-slide move). Any sequence of pebble-slide moves leading to an add-edge move can be replaced with one that has no (M2) steps and allows the same add-edge move. In other words, if it is possible to collect `+ 1 pebbles on the ends of an edge to be added, then it is possible to do this without creating any monochromatic cycles. 12 Ileana Streinu, Louis Theran Figure 7 and Figure 8 illustrate the construction used in the proof of Lemma 15. We call this the shortcut construction by analogy to matroid union and intersection augmenting paths used in previous work on the lower range. Figure 6 shows the structure of the proof. The shortcut construction removes an (M2) step at the beginning of a sequence that reorients a path from v to w with pebble-slides. Since one application of the shortcut construction reorients a simple path from a vertex w′ to w, and a path from v to w′ is preserved, the shortcut construction can be applied inductively to find the sequence of moves we want. Fig. 6. Outline of the shortcut construction: (a) An arbitrary simple path from v to w with curved lines indicating simple paths. (b) An (M2) step. The black edge, about to be flipped, would create a cycle, shown in dashed and solid gray, of the (unique) gray tree rooted at w. The solid gray edges were part of the original path from (a). (c) The shortened path to the gray pebble; the new path follows the gray tree all the way from the first time the original path touched the gray tree at w′. The path from v to w′ is simple, and the shortcut construction can be applied inductively to it. Proof. Without loss of generality, we can assume that our sequence of moves reorients a simple path in H, and that the first move (the end of the path) is (M2). The (M2) step moves a pebble of color ci from a vertex w onto the edge vw, which is reversed. Because the move is (M2), v and w are contained in a maximal monochromatic tree of color ci. Call this tree H ′i , and observe that it is rooted at w. Now consider the edges reversed in our sequence of moves. As noted above, before we make any of the moves, these sketch out a simple path in H ending at w. Let z be the first vertex on this path in H ′i . We modify our sequence of moves as follows: delete, from the beginning, every move before the one that reverses some edge yz; prepend onto what is left a sequence of moves that moves the pebble on w to z in H ′i . Sparsity-certifying Graph Decompositions 13 Fig. 7. Eliminating (M2) moves: (a) an (M2) move; (b) avoiding the (M2) by moving along another path. The path where the pebbles move is indicated by doubled lines. Fig. 8. Eliminating (M2) moves: (a) the first step to move the black pebble along the doubled path is (M2); (b) avoiding the (M2) and simplifying the path. Since no edges change color in the beginning of the new sequence, we have eliminated the (M2) move. Because our construction does not change any of the edges involved in the remaining tail of the original sequence, the part of the original path that is left in the new sequence will still be a simple path in H, meeting our initial hypothesis. The rest of the lemma follows by induction. Together Lemma 14 and Lemma 15 prove the following. Lemma 16. If G is a pebble-game graph, then G has a canonical pebble game construction. Using canonical pebble game constructions, we can identify the tight pebble-game graphs with maps-and-trees and `Tk graphs. 14 Ileana Streinu, Louis Theran Theorem 3 (Main Theorem (Lower Range): Maps-and-trees coincide with pebble-game graphs). Let 0 ≤ ` ≤ k. A graph G is a tight pebble-game graph if and only if G is a (k, `)- maps-and-trees. Proof. As observed above, a maps-and-trees decomposition is a special case of the pebble game decomposition. Applying Theorem 2, we see that any maps-and-trees must be a pebble-game graph. For the reverse direction, consider a canonical pebble game construction of a tight graph. From Lemma 8, we see that there are ` pebbles left on G at the end of the construction. The definition of the canonical add-edge move implies that there must be at least one pebble of each ci for i = 1,2, . . . , `. It follows that there is exactly one of each of these colors. By Lemma 12, each of these pebbles is the root of a monochromatic tree-piece with n− 1 edges, yielding the required ` edge-disjoint spanning trees. Corollary 5 (Nash-Williams [17], Tutte [23], White and Whiteley [24]). Let `≤ k. A graph G is tight if and only if has a (k, `)-maps-and-trees decomposition. We next consider the decompositions induced by canonical pebble game constructions when `≥ k +1. Theorem 4 (Main Theorem (Upper Range): Proper Trees-and-trees coincide with peb- ble-game graphs). Let k≤ `≤ 2k−1. A graph G is a tight pebble-game graph if and only if it is a proper `Tk with kn− ` edges. Proof. As observed above, a proper `Tk decomposition must be sparse. What we need to show is that a canonical pebble game construction of a tight graph produces a proper `Tk . By Theorem 2 and Lemma 16, we already have the condition on tree-pieces and the decom- position into ` edge-disjoint trees. Finally, an application of (I4), shows that every vertex must in in exactly k of the trees, as required. Corollary 6 (Crapo [2], Haas [7]). Let k ≤ `≤ 2k−1. A graph G is tight if and only if it is a proper `Tk . 8. Pebble game algorithms for finding decompositions A naı̈ve implementation of the constructions in the previous section leads to an algorithm re- quiring Θ(n2) time to collect each pebble in a canonical construction: in the worst case Θ(n) applications of the construction in Lemma 15 requiring Θ(n) time each, giving a total running time of Θ(n3) for the decomposition problem. In this section, we describe algorithms for the decomposition problem that run in time O(n2). We begin with the overall structure of the algorithm. Algorithm 17 (The canonical pebble game with colors). Input: A graph G. Output: A pebble-game graph H. Method: – Set V (H) = V (G) and place one pebble of each color on the vertices of H. – For each edge vw ∈ E(G) try to collect at least `+1 pebbles on v and w using pebble-slide moves as described by Lemma 15. Sparsity-certifying Graph Decompositions 15 – If at least `+1 pebbles can be collected, add vw to H using an add-edge move as in Lemma 14, otherwise discard vw. – Finally, return H, and the locations of the pebbles. Correctness. Theorem 1 and the result from [24] that the sparse graphs are the independent sets of a matroid show that H is a maximum sized sparse subgraph of G. Since the construction found is canonical, the main theorem shows that the coloring of the edges in H gives a maps- and-trees or proper `Tk decomposition. Complexity. We start by observing that the running time of Algorithm 17 is the time taken to process O(n) edges added to H and O(m) edges not added to H. We first consider the cost of an edge of G that is added to H. Each of the pebble game moves can be implemented in constant time. What remains is to describe an efficient way to find and move the pebbles. We use the following algorithm as a subroutine of Algorithm 17 to do this. Algorithm 18 (Finding a canonical path to a pebble.). Input: Vertices v and w, and a pebble game configuration on a directed graph H. Output: If a pebble was found, ‘yes’, and ‘no’ otherwise. The configuration of H is updated. Method: – Start by doing a depth-first search from from v in H. If no pebble not on w is found, stop and return ‘no.’ – Otherwise a pebble was found. We now have a path v = v1,e1, . . . ,ep−1,vp = u, where the vi are vertices and ei is the edge vivi+1. Let c[ei] be the color of the pebble on ei. We will use the array c[] to keep track of the colors of pebbles on vertices and edges after we move them and the array s[] to sketch out a canonical path from v to u by finding a successor for each edge. – Set s[u] = ‘end′ and set c[u] to the color of an arbitrary pebble on u. We walk on the path in reverse order: vp,ep−1,ep−2, . . . ,e1,v1. For each i, check to see if c[vi] is set; if so, go on to the next i. Otherwise, check to see if c[vi+1] = c[ei]. – If it is, set s[vi] = ei and set c[vi] = c[ei], and go on to the next edge. – Otherwise c[vi+1] 6= c[ei], try to find a monochromatic path in color c[vi+1] from vi to vi+1. If a vertex x is encountered for which c[x] is set, we have a path vi = x1, f1,x2, . . . , fq−1,xq = x that is monochromatic in the color of the edges; set c[xi] = c[ fi] and s[xi] = fi for i = 1,2, . . . ,q−1. If c[x] = c[ fq−1], stop. Otherwise, recursively check that there is not a monochro- matic c[x] path from xq−1 to x using this same procedure. – Finally, slide pebbles along the path from the original endpoints v to u specified by the successor array s[v], s[s[v]], . . . The correctness of Algorithm 18 comes from the fact that it is implementing the shortcut construction. Efficiency comes from the fact that instead of potentially moving the pebble back and forth, Algorithm 18 pre-computes a canonical path crossing each edge of H at most three times: once in the initial depth-first search, and twice while converting the initial path to a canonical one. It follows that each accepted edges takes O(n) time, for a total of O(n2) time spent processing edges in H. Although we have not discussed this explicity, for the algorithm to be efficient we need to maintain components as in [12]. After each accepted edge, the components of H can be updated in time O(n). Finally, the results of [12, 13] show that the rejected edges take an amortized O(1) time each. 16 Ileana Streinu, Louis Theran Summarizing, we have shown that the canonical pebble game with colors solves the decom- position problem in time O(n2). 9. An important special case: Rigidity in dimension 2 and slider-pinning In this short section we present a new application for the special case of practical importance, k = 2, ` = 3. As discussed in the introduction, Laman’s theorem [11] characterizes minimally rigid graphs as the (2,3)-tight graphs. In recent work on slider pinning, developed after the current paper was submitted, we introduced the slider-pinning model of rigidity [15, 20]. Com- binatorially, we model the bar-slider frameworks as simple graphs together with some loops placed on their vertices in such a way that there are no more than 2 loops per vertex, one of each color. We characterize the minimally rigid bar-slider graphs [20] as graphs that are: 1. (2,3)-sparse for subgraphs containing no loops. 2. (2,0)-tight when loops are included. We call these graphs (2,0,3)-graded-tight, and they are a special case of the graded-sparse graphs studied in our paper [14]. The connection with the pebble games in this paper is the following. Corollary 19 (Pebble games and slider-pinning). In any (2,3)-pebble game graph, if we replace pebbles by loops, we obtain a (2,0,3)-graded-tight graph. Proof. Follows from invariant (I3) of Lemma 7. In [15], we study a special case of slider pinning where every slider is either vertical or horizontal. We model the sliders as pre-colored loops, with the color indicating x or y direction. For this axis parallel slider case, the minimally rigid graphs are characterized by: 1. (2,3)-sparse for subgraphs containing no loops. 2. Admit a 2-coloring of the edges so that each color is a forest (i.e., has no cycles), and each monochromatic tree spans exactly one loop of its color. This also has an interpretation in terms of colored pebble games. Corollary 20 (The pebble game with colors and slider-pinning). In any canonical (2,3)- pebble-game-with-colors graph, if we replace pebbles by loops of the same color, we obtain the graph of a minimally pinned axis-parallel bar-slider framework. Proof. Follows from Theorem 4, and Lemma 12. 10. Conclusions and open problems We presented a new characterization of (k, `)-sparse graphs, the pebble game with colors, and used it to give an efficient algorithm for finding decompositions of sparse graphs into edge- disjoint trees. Our algorithm finds such sparsity-certifying decompositions in the upper range and runs in time O(n2), which is as fast as the algorithms for recognizing sparse graphs in the upper range from [12]. We also used the pebble game with colors to describe a new sparsity-certifying decomposi- tion that applies to the entire matroidal range of sparse graphs. Sparsity-certifying Graph Decompositions 17 We defined and studied a class of canonical pebble game constructions that correspond to either a maps-and-trees or proper `Tk decomposition. This gives a new proof of the Tutte-Nash- Williams arboricity theorem and a unified proof of the previously studied decomposition cer- tificates of sparsity. Canonical pebble game constructions also show the relationship between the `+1 pebble condition, which applies to the upper range of `, to matroid union augmenting paths, which do not apply in the upper range. Algorithmic consequences and open problems. In [6], Gabow and Westermann give an O(n3/2) algorithm for recognizing sparse graphs in the lower range and extracting sparse subgraphs from dense ones. Their technique is based on efficiently finding matroid union augmenting paths, which extend a maps-and-trees decomposition. The O(n3/2) algorithm uses two subroutines to find augmenting paths: cyclic scanning, which finds augmenting paths one at a time, and batch scanning, which finds groups of disjoint augmenting paths. We observe that Algorithm 17 can be used to replace cyclic scanning in Gabow and Wester- mann’s algorithm without changing the running time. The data structures used in the implemen- tation of the pebble game, detailed in [12, 13] are simpler and easier to implement than those used to support cyclic scanning. The two major open algorithmic problems related to the pebble game are then: Problem 1. Develop a pebble game algorithm with the properties of batch scanning and obtain an implementable O(n3/2) algorithm for the lower range. Problem 2. Extend batch scanning to the `+1 pebble condition and derive an O(n3/2) pebble game algorithm for the upper range. In particular, it would be of practical importance to find an implementable O(n3/2) algorithm for decompositions into edge-disjoint spanning trees. References 1. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, vol. 2832, pp. 78–89. (2003) 2. Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988) 3. Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965) 4. Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003) 5. Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50, 259–273 (1995) 6. Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992) 7. Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002) 8. Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompo- sitions. Journal of Combinatorial Mathematics and Combinatorial Computing 62, 3–11 (2007) 9. Hendrickson, B.: Conditions for unique graph realizations. SIAM Journal on Computing 21(1), 65–84 (1992) 18 Ileana Streinu, Louis Theran 10. Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics 137, 346–365 (1997) 11. Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4, 331–340 (1970) 12. Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008) 13. Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Cana- dian Conference of Computational Geometry. Windsor, Ontario (2005). http://cccg. cs.uwindsor.ca/papers/72.pdf 14. Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. Journal of Universal Computer Science 13(10) (2007) 15. Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007) 16. Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North-Holland, Amsterdam (1979) 17. Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. Journal of the London Mathematical Society 39, 12 (1964) 18. Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992) 19. Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Mathematics of Operations Research 10(4), 701–708 (1985) 20. Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Pro- ceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365– 374. ACM, New York, NY, USA (2008). 21. Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. Journal of Combinato- rial Theory, Series B 26, 95–112 (1984) 22. Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993) 23. Tutte, W.T.: On the problem of decomposing a graph into n connected factors. Journal of the London Mathematical Society 142, 221–230 (1961) 24. Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics 1(2), 237–255 (1988) http://cccg.cs.uwindsor.ca/papers/72.pdf http://cccg.cs.uwindsor.ca/papers/72.pdf Introduction and preliminaries Historical background The pebble game with colors Our Results Pebble game graphs The pebble-game-with-colors decomposition Canonical Pebble Game Constructions Pebble game algorithms for finding decompositions An important special case: Rigidity in dimension 2 and slider-pinning Conclusions and open problems

The evolution of the Earth-Moon system based on the dark fluid model The evolution of the Earth-Moon system based on the dark matter field fluid model Hongjun Pan Department of Chemistry University of North Texas, Denton, Texas 76203, U. S. A. Abstract The evolution of Earth-Moon system is described by the dark matter field fluid model with a non-Newtonian approach proposed in the Meeting of Division of Particle and Field 2004, American Physical Society. The current behavior of the Earth-Moon system agrees with this model very well and the general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, which is far beyond the Roche’s limit. The result suggests that the tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The average dark matter field fluid constant derived from Earth-Moon system data is 4.39 × 10-22 s-1m-1. This model predicts that the Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. Key Words. dark matter, fluid, evolution, Earth, Moon, Mars 1. Introduction The popularly accepted theory for the formation of the Earth-Moon system is that the Moon was formed from debris of a strong impact by a giant planetesimal with the Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the formation of the Earth-Moon system, it has been evolving at all time scale. It is well known that the Moon is receding from us and both the Earth’s rotation and Moon’s rotation are slowing. The popular theory is that the tidal friction causes all those changes based on the conservation of the angular momentum of the Earth-Moon system. The situation becomes complicated in describing the past evolution of the Earth-Moon system. Because the Moon is moving away from us and the Earth rotation is slowing, this means that the Moon was closer and the Earth rotation was faster in the past. Creationists argue that based on the tidal friction theory, the tidal friction should be stronger and the recessional rate of the Moon should be greater in the past, the distance of the Moon would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence indicates that the recession of the Moon in the past was slower than the present rate, i. e., the recession has been accelerating with time. Therefore, it must be concluded that tidal friction was very much less in the remote past than we would deduce on the basis of present-day observations (Stacey 1977). This was called “geological time scale difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the tidal friction theory (Brush 1983). But we have to consider the case carefully in various aspects. One possible scenario is that the Earth has been undergoing dynamic evolution at all time scale since its inception, the geological and physical conditions (such as the continent positions and drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at remote past could be substantially different from currently, in which the tidal friction could be much less; therefore, the receding rate of the Moon could be slower. Various tidal friction models were proposed in the past to describe the evolution of the Earth- Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present rate of tidal dissipation is anomalously high because the tidal force is close to a resonance in the response function of ocean (Brush 1983). Kagan gave a detailed review about those tidal friction models (Kagan 1997). Those models are based on many assumptions about geological (continental position and drifting) and physical conditions in the past, and many parameters (such as phase lag angle, multi-mode approximation with time dependent frequencies of the resonance modes, etc.) have to be introduced and carefully adjusted to make their predictions close to the geological evidence. However, those assumptions and parameters are still challenged, to certain extent, as concoction. The second possible scenario is that another mechanism could dominate the evolution of the Earth-Moon system and the role of the tidal friction is not significant. In the Meeting of Division of Particle and Field 2004, American Physical Society, University of California at Riverside, the author proposed a dark matter field fluid model (Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with this model very well. This paper will demonstrate that the past evolution of Moon-Earth system can be described by the dark matter field fluid model without any assumptions about past geological and physical conditions. Although the subject of the evolution of the Earth-Moon system has been extensively studied analytically or numerically, to the author’s knowledge, there are no theories similar or equivalent to this model. 2. Invisible matter In modern cosmology, it was proposed that the visible matter in the universe is about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently invisible which is called dark matter and dark energy, such invisible matter has an anti- gravity property to make the universe expanding faster and faster. If the ratio of the matter components of the universe is close to this hypothesis, then, the evolution of the universe should be dominated by the physical mechanism of such invisible matter, such physical mechanism could be far beyond the current Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian physics could reflect only a corner of the iceberg of the greater physics. If the ratio of the matter components of the universe is close to this hypothesis, then, it should be more reasonable to think that such dominant invisible matter spreads in everywhere of the universe (the density of the invisible matter may vary from place to place); in other words, all visible matter objects should be surrounded by such invisible matter and the motion of the visible matter objects should be affected by the invisible matter if there are interactions between the visible matter and the invisible matter. If the ratio of the matter components of the universe is close to this hypothesis, then, the size of the particles of the invisible matter should be very small and below the detection limit of the current technology; otherwise, it would be detected long time ago with such dominant amount. With such invisible matter in mind, we move to the next section to develop the dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible matter (dark matter, dark energy and possible other terms) is called dark matter here. 3. The dark matter field fluid model In this proposed model, it is assumed that: 1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly filled with the dark matter which is in quiescent state relative to the motion of the celestial body. The dark matter possesses a field property and a fluid property; it can interact with the celestial body with its fluid and field properties; therefore, it can have energy exchange with the celestial body, and affect the motion of the celestial body. 2. The fluid property follows the general principle of fluid mechanics. The dark matter field fluid particles may be so small that they can easily permeate into ordinary “baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the manner of a sponge moving thru water. The nature of the field property of the dark matter field fluid is unknown. It is here assumed that the interaction of the field associated with the dark matter field fluid with the celestial body is proportional to the mass of the celestial body. The dark matter field fluid is assumed to have a repulsive force against the gravitational force towards baryonic matter. The nature and mechanism of such repulsive force is unknown. With the assumptions above, one can study how the dark matter field fluid may influence the motion of a celestial body and compare the results with observations. The common shape of celestial bodies is spherical. According to Stokes's law, a rigid non- permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds number experiences a resistance force F rvF πμ6−= (1) where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual properties of the dark matter field fluid and its permeation into the sphere, the force F may not be proportional to the radius of the sphere. Also, F may be proportional to the mass of the sphere due to the field interaction. Therefore, with the combined effects of both fluid and field, the force exerted on the sphere by the dark matter field fluid is assumed to be of the scaled form (2) mvrF n−−= 16πη where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η is the dark matter field fluid constant, which is equivalent to μ. The direction of the resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force described by Eq. 2 is velocity-dependent and causes negative acceleration. According to Newton's second law of motion, the equation of motion for the sphere is mvr m n−−= 16πη (3) Then (4) )6exp( 10 vtrvv n−−= πη where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a massive gravitational center, there are three forces in the line between the sphere and the gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and (3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the orbital velocity and causes the sphere to move inward to the gravitational center. However, if the sum of the centripetal acceleration force and the repulsive force is stronger than the gravitational force, then, the sphere will move outward and recede from the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is sufficiently slow and the repulsive force is small compared to the gravitational force and centripetal acceleration force, then the rate of receding will be accordingly relatively slow. Therefore, the gravitational force and the centripetal acceleration force can be approximately treated in equilibrium at any time. The pseudo equilibrium equation is GMm 2 2 = (5) where G is the gravitational constant, M is the mass of the gravitational center, and R is the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields )12exp( 1 R n−= πη (6) (7) )12exp( 10 trRR n−= πη where R = (8) R0 is the initial distance to the gravitational center. Note that R exponentially increases with time. The increase of orbital energy with the receding comes from the repulsive force of dark matter field fluid. The recessional rate of the sphere is dR n−= 112πη (9) The acceleration of the recession is ( Rr Rd n 21 12 −= πη ) . (10) The recessional acceleration is positive and proportional to its distance to the gravitational center, so the recession is faster and faster. According to the mechanics of fluids, for a rigid non-permeable sphere rotating about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere ωπμ 38 rT −= (11) where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted on the sphere is ( ) ωπη mrT n 318 −−= (12) The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The torque causes the negative angular acceleration = (13) where I is the moment of inertia of the sphere in the dark matter field fluid ( )21 2 nrmI −= (14) Therefore, the equation of rotation for the sphere in the dark matter field fluid is ωπη d −−= 120 (15) Solving this equation yields (16) )20exp( 10 tr n−−= πηωω where ω0 is the initial angular velocity. One can see that the angular velocity of the sphere exponentially decreases with time and the angular deceleration is proportional to its angular velocity. For the same celestial sphere, combining Eq. 9 and Eq. 15 yields (17) The significance of Eq. 17 is that it contains only observed data without assumptions and undetermined parameters; therefore, it is a critical test for this model. For two different celestial spheres in the same system, combining Eq. 9 and Eq. 15 yields 67.1 1 −=−=⎟⎟ (18) This is another critical test for this model. 4. The current behavior of the Earth-Moon system agrees with the model The Moon-Earth system is the simplest gravitational system. The solar system is complex, the Earth and the Moon experience not only the interaction of the Sun but also interactions of other planets. Let us consider the local Earth-Moon gravitational system as an isolated local gravitational system, i.e., the influence from the Sun and other planets on the rotation and orbital motion of the Moon and on the rotation of the Earth is assumed negligible compared to the forces exerted by the moon and earth on each other. In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are listed below for the readers' convenience to verify the calculation because the data may vary slightly with different data sources. Moon: Mean radius: r = 1738.0 km Mass: m = 7.3483 × 1025 gram Rotation period = 27.321661 days Angular velocity of Moon = 2.6617 × 10-6 rad s-1 Mean distance to Earth Rm= 384400 km Mean orbital velocity v = 1.023 km s-1 Orbit eccentricity e = 0.0549 Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2 = (-1.255 ± 0.024) × 10-4 rad century-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1 Earth: Mean radius: r = 6371.0 km Mass: m = 5.9742 × 1027 gram Rotation period = 23 h 56m 04.098904s = 86164.098904s Angular velocity of rotation = 7.292115 × 10-5 rad s-1 Mean distance to the Sun Rm= 149,597,870.61 km Mean orbital velocity v = 29.78 km s-1 Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2 The Moon's angular rotation acceleration rate and increase in mean distance to the Earth (receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the result is 039.054.1 10662.21021.1 1092509.31026.1 (19) The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord with the model used here. The difference (about 7.8%) between the values of -1.54 and - 1.67 may come from several sources: 1. Moon's orbital is not a perfect circle 2. Moon is not a perfect rigid sphere. 3. The effect from Sun and other planets. 4. Errors in data. 5. Possible other unknown reasons. The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data sets. The third data set can be used to further test the model. If this model correctly describes the situation at hand, it should give consistent results for different motions. The values of n and η calculated from three different data sets are listed below (Note, the mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are used in the calculation). The value of n: n = 0.64 From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1 From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1 From the Moon's recession: η = 4.64 × 10-22 s-1 m-1 One can see that the three values of η are consistent within the range of error in the data. The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1 By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into Eq. 18, the result is 14.053.1 6371000 1738000 1021.11029.7 1092509.3105.5 )64.01( (20) This is also in accord with the model used here. The dragging force exerted on the Moon's orbital motion by the dark matter field fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, respectively. 5. The evolution of Earth-Moon system Sonett et al. found that the length of the terrestrial day 900 million years ago was about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 1996). According to the model presented here, back in that time, the length of the day was about 19.2 hours, this agrees very well with Sonett et al.'s result. Another critical aspect of modeling the evolution of the Earth-Moon system is to give a reasonable estimate of the closest distance of the Moon to the Earth when the system was established at 4.5 billion years ago. Based on the dark matter field fluid model, and the above result, the closest distance of the Moon to the Earth was about 259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion years ago was 250000 km, this is probably the most reasonable number that most astronomers believe and it agrees excellently with the result of this model. The closest distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 242000 km (Hansen, 1982). According to this model, the length of day of the Earth was about 8 hours at 4.5 billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the length of day of the Earth with the age of the Earth-Moon system described by this model along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One can see that those data fit this model very well in their time range. Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model for past 900 million years. One can see that the model agrees with the geological and fossil data beautifully. The important difference of this model with early models in describing the early evolution of the Earth-Moon system is that this model is based only on current data of the Moon-Earth system and there are no assumptions about the conditions of earlier Earth rotation and continental drifting. Based on this model, the Earth-Moon system has been smoothly evolving to the current position since it was established and the recessional rate of the Moon has been gradually increasing, however, this description does not take it into account that there might be special events happened in the past to cause the suddenly significant changes in the motions of the Earth and the Moon, such as strong impacts by giant asteroids and comets, etc, because those impacts are very common in the universe. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially increases with time. One may then imagine that the recessional rate will quickly become very large. The increase is in fact extremely slow. The Moon's recessional rate will be 3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. However, whether the Moon's recession will continue or at some time later another mechanism will take over is not known. It should be understood that the tidal friction does affect the evolution of the Earth itself such as the surface crust structure, continental drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s rotation, however, such role is not a dominant mechanism. Unfortunately, there is no data available for the changes of the Earth's orbital motion and all other members of solar system. According to this model and above results, the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km century-1, the length of a year increases about 6.8 ms and the change of the temperature is -1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on the Earth. The length of a year at 1 billion years ago would be 80% of the current length of the year. However, much evidence (growth-bands of corals and shellfish as well as some other evidences) suggest that there has been no apparent change in the length of the year over the billion years and the Earth's orbital motion is more stable than its rotation. This suggests that dark matter field fluid is circulating around Sun with the same direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the Earth's orbital motion experiences very little or no dragging force from the dark matter field fluid. However, this is a conjecture, extensive research has to be conducted to verify if this is the case. 6. Skeptical description of the evolution of the Mars The Moon does not have liquid fluid on its surface, even there is no air, therefore, there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees with the model very well. Based on this, one may reasonably think that the Mars’s rotation should be slowing also. The Mars is our nearest neighbor which has attracted human’s great attention since ancient time. The exploration of the Mars has been heating up in recent decades. NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect data and study this mysterious planet. So far there is still not enough data about the history of this planet to describe its evolution. Same as the Earth, the Mars rotates about its central axis and revolves around the Sun, however, the Mars does not have a massive moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to slow its rotation by tidal friction theories. Based on the above result and current the Mars's data, this model predicts that the angular acceleration of the Mars should be about -4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the solar days/Mars year, the vertical dash line marks the current age of the Mars with assumption that the Mars was formed in a similar time period of the Earth formation. Such description was not given before according to the author’s knowledge and is completely skeptical due to lack of reliable data. However, with further expansion of the research and exploration on the Mars, we shall feel confident that the reliable data about the angular rotation acceleration of the Mars will be available in the near future which will provide a vital test for the prediction of this model. There are also other factors which may affect the Mars’s rotation rate such as mass redistribution due to season change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be carefully analyzed. 7. Discussion about the model From the above results, one can see that the current Earth-Moon data and the geological and fossil data agree with the model very well and the past evolution of the Earth-Moon system can be described by the model without introducing any additional parameters; this model reveals the interesting relationship between the rotation and receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in the same gravitational system, such relationship is not known before. Such success can not be explained by “coincidence” or “luck” because of so many data involved (current Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a “ad hoc” or a wrong model, although the chance for the natural happening of such “coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s data will clarify this; otherwise, a new theory from different approach can be developed to give the same or better description as this model does. It is certain that this model is not perfect and may have defects, further development may be conducted. James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity. It extends unbroken from star to star ….” The medium that Maxwell talked about is the aether which was proposed as the carrier of light wave propagation. The Michelson-Morley experiment only proved that the light wave propagation does not depend on such medium and did not reject the existence of the medium in the interstellar space. In fact, the concept of the interstellar medium has been developed dramatically recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 7. Conclusion The evolution of the Earth-Moon system can be described by the dark matter field fluid model with non-Newtonian approach and the current data of the Earth and the Moon fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of day was about 8 hours. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. References S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century: Creationist Arguments for a young Earth. Scientists confront creationism. W. W. Norton & Company, New York, London, pp 49. E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Science, 265, 482. D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J. Sediment. Res. 69(6), 1154. K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York. H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the Earth’s Rotation. Springer-Verlag, Berlin, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100. F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons. J. W. Wells, 1963. Nature, 197, 948. Caption Figure 1, the evolution of Moon’s distance and the length of day of the earth with the age of the Earth-Moon system. Solid lines are calculated according to the dark matter field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al. The dash line marks the current age of the Earth-Moon system. Figure 2, the evolution of Solar days of year with the age of the Earth-Moon system. The solid line is calculated according to dark matter field fluid model. The data are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current age (4.5 billion years). Figure 3, the skeptical description of the evolution of Mars’s length of day and the solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 billion years). The vertical dash line marks the current age of Mars. Figure 1, Moon's distance and the length of day of Earth change with the age of Earth-Moon system The age of Earth-Moon system (109 years) 0 1 2 3 4 5 Distance Length of day Roche's limit Hansen's result Figure 2, the solar days / year vs. the age of the Earth The age of the Earth (109 years) 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6

The evolution of Earth-Moon system is described by the dark matter field fluid model proposed in the Meeting of Division of Particle and Field 2004, American Physical Society. The current behavior of the Earth-Moon system agrees with this model very well and the general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The closest distance of the Moon to Earth was about 259000 km at 4.5 billion years ago, which is far beyond the Roche's limit. The result suggests that the tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The average dark matter field fluid constant derived from Earth-Moon system data is 4.39 x 10^(-22) s^(-1)m^(-1). This model predicts that the Mars's rotation is also slowing with the angular acceleration rate about -4.38 x 10^(-22) rad s^(-2).

Introduction The popularly accepted theory for the formation of the Earth-Moon system is that the Moon was formed from debris of a strong impact by a giant planetesimal with the Earth at the close of the planet-forming period (Hartmann and Davis 1975). Since the formation of the Earth-Moon system, it has been evolving at all time scale. It is well known that the Moon is receding from us and both the Earth’s rotation and Moon’s rotation are slowing. The popular theory is that the tidal friction causes all those changes based on the conservation of the angular momentum of the Earth-Moon system. The situation becomes complicated in describing the past evolution of the Earth-Moon system. Because the Moon is moving away from us and the Earth rotation is slowing, this means that the Moon was closer and the Earth rotation was faster in the past. Creationists argue that based on the tidal friction theory, the tidal friction should be stronger and the recessional rate of the Moon should be greater in the past, the distance of the Moon would quickly fall inside the Roche's limit (for earth, 15500 km) in which the Moon would be torn apart by gravity in 1 to 2 billion years ago. However, geological evidence indicates that the recession of the Moon in the past was slower than the present rate, i. e., the recession has been accelerating with time. Therefore, it must be concluded that tidal friction was very much less in the remote past than we would deduce on the basis of present-day observations (Stacey 1977). This was called “geological time scale difficulty” or “Lunar crisis” and is one of the main arguments by creationists against the tidal friction theory (Brush 1983). But we have to consider the case carefully in various aspects. One possible scenario is that the Earth has been undergoing dynamic evolution at all time scale since its inception, the geological and physical conditions (such as the continent positions and drifting, the crust, surface temperature fluctuation like the glacial/snowball effect, etc) at remote past could be substantially different from currently, in which the tidal friction could be much less; therefore, the receding rate of the Moon could be slower. Various tidal friction models were proposed in the past to describe the evolution of the Earth- Moon system to avoid such difficulty or crisis and put the Moon at quite a comfortable distance from Earth at 4.5 billion years ago (Hansen 1982, Kagan and Maslova 1994, Ray et al. 1999, Finch 1981, Slichter 1963). The tidal friction theories explain that the present rate of tidal dissipation is anomalously high because the tidal force is close to a resonance in the response function of ocean (Brush 1983). Kagan gave a detailed review about those tidal friction models (Kagan 1997). Those models are based on many assumptions about geological (continental position and drifting) and physical conditions in the past, and many parameters (such as phase lag angle, multi-mode approximation with time dependent frequencies of the resonance modes, etc.) have to be introduced and carefully adjusted to make their predictions close to the geological evidence. However, those assumptions and parameters are still challenged, to certain extent, as concoction. The second possible scenario is that another mechanism could dominate the evolution of the Earth-Moon system and the role of the tidal friction is not significant. In the Meeting of Division of Particle and Field 2004, American Physical Society, University of California at Riverside, the author proposed a dark matter field fluid model (Pan 2005) with a non-Newtonian approach, the current Moon and Earth data agree with this model very well. This paper will demonstrate that the past evolution of Moon-Earth system can be described by the dark matter field fluid model without any assumptions about past geological and physical conditions. Although the subject of the evolution of the Earth-Moon system has been extensively studied analytically or numerically, to the author’s knowledge, there are no theories similar or equivalent to this model. 2. Invisible matter In modern cosmology, it was proposed that the visible matter in the universe is about 2 ~ 10 % of the total matter and about 90 ~ 98% of total matter is currently invisible which is called dark matter and dark energy, such invisible matter has an anti- gravity property to make the universe expanding faster and faster. If the ratio of the matter components of the universe is close to this hypothesis, then, the evolution of the universe should be dominated by the physical mechanism of such invisible matter, such physical mechanism could be far beyond the current Newtonian physics and Einsteinian physics, and the Newtonian physics and Einsteinian physics could reflect only a corner of the iceberg of the greater physics. If the ratio of the matter components of the universe is close to this hypothesis, then, it should be more reasonable to think that such dominant invisible matter spreads in everywhere of the universe (the density of the invisible matter may vary from place to place); in other words, all visible matter objects should be surrounded by such invisible matter and the motion of the visible matter objects should be affected by the invisible matter if there are interactions between the visible matter and the invisible matter. If the ratio of the matter components of the universe is close to this hypothesis, then, the size of the particles of the invisible matter should be very small and below the detection limit of the current technology; otherwise, it would be detected long time ago with such dominant amount. With such invisible matter in mind, we move to the next section to develop the dark matter field fluid model with non-Newtonian approach. For simplicity, all invisible matter (dark matter, dark energy and possible other terms) is called dark matter here. 3. The dark matter field fluid model In this proposed model, it is assumed that: 1. A celestial body rotates and moves in the space, which, for simplicity, is uniformly filled with the dark matter which is in quiescent state relative to the motion of the celestial body. The dark matter possesses a field property and a fluid property; it can interact with the celestial body with its fluid and field properties; therefore, it can have energy exchange with the celestial body, and affect the motion of the celestial body. 2. The fluid property follows the general principle of fluid mechanics. The dark matter field fluid particles may be so small that they can easily permeate into ordinary “baryonic” matter; i. e., ordinary matter objects could be saturated with such dark matter field fluid. Thus, the whole celestial body interacts with the dark matter field fluid, in the manner of a sponge moving thru water. The nature of the field property of the dark matter field fluid is unknown. It is here assumed that the interaction of the field associated with the dark matter field fluid with the celestial body is proportional to the mass of the celestial body. The dark matter field fluid is assumed to have a repulsive force against the gravitational force towards baryonic matter. The nature and mechanism of such repulsive force is unknown. With the assumptions above, one can study how the dark matter field fluid may influence the motion of a celestial body and compare the results with observations. The common shape of celestial bodies is spherical. According to Stokes's law, a rigid non- permeable sphere moving through a quiescent fluid with a sufficiently low Reynolds number experiences a resistance force F rvF πμ6−= (1) where v is the moving velocity, r is the radius of the sphere, and μ is the fluid viscosity constant. The direction of the resistance force F in Eq. 1 is opposite to the direction of the velocity v. For a rigid sphere moving through the dark matter field fluid, due to the dual properties of the dark matter field fluid and its permeation into the sphere, the force F may not be proportional to the radius of the sphere. Also, F may be proportional to the mass of the sphere due to the field interaction. Therefore, with the combined effects of both fluid and field, the force exerted on the sphere by the dark matter field fluid is assumed to be of the scaled form (2) mvrF n−−= 16πη where n is a parameter arising from saturation by dark matter field fluid, the r1-n can be viewed as the effective radius with the same unit as r, m is the mass of the sphere, and η is the dark matter field fluid constant, which is equivalent to μ. The direction of the resistance force F in Eq. 2 is opposite to the direction of the velocity v. The force described by Eq. 2 is velocity-dependent and causes negative acceleration. According to Newton's second law of motion, the equation of motion for the sphere is mvr m n−−= 16πη (3) Then (4) )6exp( 10 vtrvv n−−= πη where v0 is the initial velocity (t = 0) of the sphere. If the sphere revolves around a massive gravitational center, there are three forces in the line between the sphere and the gravitational center: (1) the gravitational force, (2) the centripetal acceleration force; and (3) the repulsive force of the dark matter field fluid. The drag force in Eq. 3 reduces the orbital velocity and causes the sphere to move inward to the gravitational center. However, if the sum of the centripetal acceleration force and the repulsive force is stronger than the gravitational force, then, the sphere will move outward and recede from the gravitational center. This is the case of interest here. If the velocity change in Eq. 3 is sufficiently slow and the repulsive force is small compared to the gravitational force and centripetal acceleration force, then the rate of receding will be accordingly relatively slow. Therefore, the gravitational force and the centripetal acceleration force can be approximately treated in equilibrium at any time. The pseudo equilibrium equation is GMm 2 2 = (5) where G is the gravitational constant, M is the mass of the gravitational center, and R is the radius of the orbit. Inserting v of Eq. 4 into Eq. 5 yields )12exp( 1 R n−= πη (6) (7) )12exp( 10 trRR n−= πη where R = (8) R0 is the initial distance to the gravitational center. Note that R exponentially increases with time. The increase of orbital energy with the receding comes from the repulsive force of dark matter field fluid. The recessional rate of the sphere is dR n−= 112πη (9) The acceleration of the recession is ( Rr Rd n 21 12 −= πη ) . (10) The recessional acceleration is positive and proportional to its distance to the gravitational center, so the recession is faster and faster. According to the mechanics of fluids, for a rigid non-permeable sphere rotating about its central axis in the quiescent fluid, the torque T exerted by the fluid on the sphere ωπμ 38 rT −= (11) where ω is the angular velocity of the sphere. The direction of the torque in Eq. 11 is opposite to the direction of the rotation. In the case of a sphere rotating in the quiescent dark matter field fluid with angular velocity ω, similar to Eq. 2, the proposed T exerted on the sphere is ( ) ωπη mrT n 318 −−= (12) The direction of the torque in Eq. 12 is opposite to the direction of the rotation. The torque causes the negative angular acceleration = (13) where I is the moment of inertia of the sphere in the dark matter field fluid ( )21 2 nrmI −= (14) Therefore, the equation of rotation for the sphere in the dark matter field fluid is ωπη d −−= 120 (15) Solving this equation yields (16) )20exp( 10 tr n−−= πηωω where ω0 is the initial angular velocity. One can see that the angular velocity of the sphere exponentially decreases with time and the angular deceleration is proportional to its angular velocity. For the same celestial sphere, combining Eq. 9 and Eq. 15 yields (17) The significance of Eq. 17 is that it contains only observed data without assumptions and undetermined parameters; therefore, it is a critical test for this model. For two different celestial spheres in the same system, combining Eq. 9 and Eq. 15 yields 67.1 1 −=−=⎟⎟ (18) This is another critical test for this model. 4. The current behavior of the Earth-Moon system agrees with the model The Moon-Earth system is the simplest gravitational system. The solar system is complex, the Earth and the Moon experience not only the interaction of the Sun but also interactions of other planets. Let us consider the local Earth-Moon gravitational system as an isolated local gravitational system, i.e., the influence from the Sun and other planets on the rotation and orbital motion of the Moon and on the rotation of the Earth is assumed negligible compared to the forces exerted by the moon and earth on each other. In addition, the eccentricity of the Moon's orbit is small enough to be ignored. The data about the Moon and the Earth from references (Dickey et .al., 1994, and Lang, 1992) are listed below for the readers' convenience to verify the calculation because the data may vary slightly with different data sources. Moon: Mean radius: r = 1738.0 km Mass: m = 7.3483 × 1025 gram Rotation period = 27.321661 days Angular velocity of Moon = 2.6617 × 10-6 rad s-1 Mean distance to Earth Rm= 384400 km Mean orbital velocity v = 1.023 km s-1 Orbit eccentricity e = 0.0549 Angular rotation acceleration rate = -25.88 ± 0.5 arcsec century-2 = (-1.255 ± 0.024) × 10-4 rad century-2 = (-1.260 ± 0.024) × 10-23 rad s-2 Receding rate from Earth = 3.82 ± 0.07 cm year-1 = (1.21 ± 0.02) × 10-9 m s-1 Earth: Mean radius: r = 6371.0 km Mass: m = 5.9742 × 1027 gram Rotation period = 23 h 56m 04.098904s = 86164.098904s Angular velocity of rotation = 7.292115 × 10-5 rad s-1 Mean distance to the Sun Rm= 149,597,870.61 km Mean orbital velocity v = 29.78 km s-1 Angular acceleration of Earth = (-5.5 ± 0.5) × 10-22 rad s-2 The Moon's angular rotation acceleration rate and increase in mean distance to the Earth (receding rate) were obtained from the lunar laser ranging of the Apollo Program (Dickey et .al., 1994). By inserting the data of the Moon's rotation and recession into Eq. 17, the result is 039.054.1 10662.21021.1 1092509.31026.1 (19) The distance R in Eq. 19 is from the Moon's center to the Earth's center and the number 384400 km is assumed to be the distance from the Moon's surface to the Earth's surface. Eq. 19 is in good agreement with the theoretical value of -1.67. The result is in accord with the model used here. The difference (about 7.8%) between the values of -1.54 and - 1.67 may come from several sources: 1. Moon's orbital is not a perfect circle 2. Moon is not a perfect rigid sphere. 3. The effect from Sun and other planets. 4. Errors in data. 5. Possible other unknown reasons. The two parameters n and η in Eq. 9 and Eq. 15 can be determined with two data sets. The third data set can be used to further test the model. If this model correctly describes the situation at hand, it should give consistent results for different motions. The values of n and η calculated from three different data sets are listed below (Note, the mean distance of the Moon to the Earth and mean radii of the Moon and the Earth are used in the calculation). The value of n: n = 0.64 From the Moon's rotation: η = 4.27 × 10-22 s-1 m-1 From the Earth's rotation: η = 4.26 × 10-22 s-1 m-1 From the Moon's recession: η = 4.64 × 10-22 s-1 m-1 One can see that the three values of η are consistent within the range of error in the data. The average value of η: η = (4.39 ± 0.22) × 10-22 s-1 m-1 By inserting the data of the Earth's rotation, the Moon’s recession and the value of n into Eq. 18, the result is 14.053.1 6371000 1738000 1021.11029.7 1092509.3105.5 )64.01( (20) This is also in accord with the model used here. The dragging force exerted on the Moon's orbital motion by the dark matter field fluid is -1.11 × 108 N, this is negligibly small compared to the gravitational force between the Moon and the Earth ~ 1.90 × 1020 N; and the torque exerted by the dark matter field fluid on the Earth’s and the Moon's rotations is T = -5.49 × 1016 Nm and -1.15 × 1012 Nm, respectively. 5. The evolution of Earth-Moon system Sonett et al. found that the length of the terrestrial day 900 million years ago was about 19.2 hours based on the laminated tidal sediments on the Earth (Sonett et al., 1996). According to the model presented here, back in that time, the length of the day was about 19.2 hours, this agrees very well with Sonett et al.'s result. Another critical aspect of modeling the evolution of the Earth-Moon system is to give a reasonable estimate of the closest distance of the Moon to the Earth when the system was established at 4.5 billion years ago. Based on the dark matter field fluid model, and the above result, the closest distance of the Moon to the Earth was about 259000 km (center to center) or 250900 km (surface to surface) at 4.5 billion years ago, this is far beyond the Roche's limit. In the modern astronomy textbook by Chaisson and McMillan (Chaisson and McMillan, 1993, p.173), the estimated distance at 4.5 billion years ago was 250000 km, this is probably the most reasonable number that most astronomers believe and it agrees excellently with the result of this model. The closest distance of the Moon to the Earth by Hansen’s models was about 38 Earth radii or 242000 km (Hansen, 1982). According to this model, the length of day of the Earth was about 8 hours at 4.5 billion years ago. Fig. 1 shows the evolution of the distance of Moon to the Earth and the length of day of the Earth with the age of the Earth-Moon system described by this model along with data from Kvale et al. (1999), Sonett et al. (1996) and Scrutton (1978). One can see that those data fit this model very well in their time range. Fig. 2 shows the geological data of solar days year-1 from Wells (1963) and from Sonett et al. (1996) and the description (solid line) by this dark matter field fluid model for past 900 million years. One can see that the model agrees with the geological and fossil data beautifully. The important difference of this model with early models in describing the early evolution of the Earth-Moon system is that this model is based only on current data of the Moon-Earth system and there are no assumptions about the conditions of earlier Earth rotation and continental drifting. Based on this model, the Earth-Moon system has been smoothly evolving to the current position since it was established and the recessional rate of the Moon has been gradually increasing, however, this description does not take it into account that there might be special events happened in the past to cause the suddenly significant changes in the motions of the Earth and the Moon, such as strong impacts by giant asteroids and comets, etc, because those impacts are very common in the universe. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological evidence. Based on Eq. 9, the recessional rate exponentially increases with time. One may then imagine that the recessional rate will quickly become very large. The increase is in fact extremely slow. The Moon's recessional rate will be 3.04 × 10-9 m s-1 after 10 billion years and 7.64 × 10-9 m s-1 after 20 billion years. However, whether the Moon's recession will continue or at some time later another mechanism will take over is not known. It should be understood that the tidal friction does affect the evolution of the Earth itself such as the surface crust structure, continental drifting and evolution of bio-system, etc; it may also play a role in slowing the Earth’s rotation, however, such role is not a dominant mechanism. Unfortunately, there is no data available for the changes of the Earth's orbital motion and all other members of solar system. According to this model and above results, the recessional rate of the Earth should be 6.86 × 10-7 m s-1 = 21.6 m year-1 = 2.16 km century-1, the length of a year increases about 6.8 ms and the change of the temperature is -1.8 × 10-8 K year-1 with constant radiation level of the Sun and the stable environment on the Earth. The length of a year at 1 billion years ago would be 80% of the current length of the year. However, much evidence (growth-bands of corals and shellfish as well as some other evidences) suggest that there has been no apparent change in the length of the year over the billion years and the Earth's orbital motion is more stable than its rotation. This suggests that dark matter field fluid is circulating around Sun with the same direction and similar speed of Earth (at least in the Earth's orbital range). Therefore, the Earth's orbital motion experiences very little or no dragging force from the dark matter field fluid. However, this is a conjecture, extensive research has to be conducted to verify if this is the case. 6. Skeptical description of the evolution of the Mars The Moon does not have liquid fluid on its surface, even there is no air, therefore, there is no ocean-like tidal friction force to slow its rotation; however, the rotation of the Moon is still slowing at significant rate of (-1.260 ± 0.024) × 10-23 rad s-2, which agrees with the model very well. Based on this, one may reasonably think that the Mars’s rotation should be slowing also. The Mars is our nearest neighbor which has attracted human’s great attention since ancient time. The exploration of the Mars has been heating up in recent decades. NASA, Russian and Europe Space Agency sent many space crafts to the Mars to collect data and study this mysterious planet. So far there is still not enough data about the history of this planet to describe its evolution. Same as the Earth, the Mars rotates about its central axis and revolves around the Sun, however, the Mars does not have a massive moon circulating it (Mars has two small satellites: Phobos and Deimos) and there is no liquid fluid on its surface, therefore, there is no apparent ocean-like tidal friction force to slow its rotation by tidal friction theories. Based on the above result and current the Mars's data, this model predicts that the angular acceleration of the Mars should be about -4.38 × 10-22 rad s-2. Figure 3 describes the possible evolution of the length of day and the solar days/Mars year, the vertical dash line marks the current age of the Mars with assumption that the Mars was formed in a similar time period of the Earth formation. Such description was not given before according to the author’s knowledge and is completely skeptical due to lack of reliable data. However, with further expansion of the research and exploration on the Mars, we shall feel confident that the reliable data about the angular rotation acceleration of the Mars will be available in the near future which will provide a vital test for the prediction of this model. There are also other factors which may affect the Mars’s rotation rate such as mass redistribution due to season change, winds, possible volcano eruptions and Mars quakes. Therefore, the data has to be carefully analyzed. 7. Discussion about the model From the above results, one can see that the current Earth-Moon data and the geological and fossil data agree with the model very well and the past evolution of the Earth-Moon system can be described by the model without introducing any additional parameters; this model reveals the interesting relationship between the rotation and receding (Eq. 17 and Eq. 18) of the same celestial body or different celestial bodies in the same gravitational system, such relationship is not known before. Such success can not be explained by “coincidence” or “luck” because of so many data involved (current Earth’s and Moon’s data and geological and fossil data) if one thinks that this is just a “ad hoc” or a wrong model, although the chance for the natural happening of such “coincidence” or “luck” could be greater than wining a jackpot lottery; the future Mars’s data will clarify this; otherwise, a new theory from different approach can be developed to give the same or better description as this model does. It is certain that this model is not perfect and may have defects, further development may be conducted. James Clark Maxwell said in the 1873 “ The vast interplanetary and interstellar regions will no longer be regarded as waste places in the universe, which the Creator has not seen fit to fill with the symbols of the manifold order of His kingdom. We shall find them to be already full of this wonderful medium; so full, that no human power can remove it from the smallest portion of space, or produce the slightest flaw in its infinite continuity. It extends unbroken from star to star ….” The medium that Maxwell talked about is the aether which was proposed as the carrier of light wave propagation. The Michelson-Morley experiment only proved that the light wave propagation does not depend on such medium and did not reject the existence of the medium in the interstellar space. In fact, the concept of the interstellar medium has been developed dramatically recently such as the dark matter, dark energy, cosmic fluid, etc. The dark matter field fluid is just a part of such wonderful medium and “precisely” described by Maxwell. 7. Conclusion The evolution of the Earth-Moon system can be described by the dark matter field fluid model with non-Newtonian approach and the current data of the Earth and the Moon fits this model very well. At 4.5 billion years ago, the closest distance of the Moon to the Earth could be about 259000 km, which is far beyond the Roche’s limit and the length of day was about 8 hours. The general pattern of the evolution of the Moon-Earth system described by this model agrees with geological and fossil evidence. The tidal friction may not be the primary cause for the evolution of the Earth-Moon system. The Mars’s rotation is also slowing with the angular acceleration rate about -4.38 × 10-22 rad s-2. References S. G. Brush, 1983. L. R. Godfrey (editor), Ghost from the Nineteenth century: Creationist Arguments for a young Earth. Scientists confront creationism. W. W. Norton & Company, New York, London, pp 49. E. Chaisson and S. McMillan. 1993. Astronomy Today, Prentice Hall, Englewood Cliffs, NJ 07632. J. O. Dickey, et al., 1994. Science, 265, 482. D. G. Finch, 1981. Earth, Moon, and Planets, 26(1), 109. K. S. Hansen, 1982. Rev. Geophys. and Space Phys. 20(3), 457. W. K. Hartmann, D. R. Davis, 1975. Icarus, 24, 504. B. A. Kagan, N. B. Maslova, 1994. Earth, Moon and Planets 66, 173. B. A. Kagan, 1997. Prog. Oceanog. 40, 109. E. P. Kvale, H. W. Johnson, C. O. Sonett, A. W. Archer, and A. Zawistoski, 1999, J. Sediment. Res. 69(6), 1154. K. Lang, 1992. Astrophysical Data: Planets and Stars, Springer-Verlag, New York. H. Pan, 2005. Internat. J. Modern Phys. A, 20(14), 3135. R. D. Ray, B. G. Bills, B. F. Chao, 1999. J. Geophys. Res. 104(B8), 17653. C. T. Scrutton, 1978. P. Brosche, J. Sundermann, (Editors.), Tidal Friction and the Earth’s Rotation. Springer-Verlag, Berlin, pp. 154. L. B. Slichter, 1963. J. Geophys. Res. 68, 14. C. P. Sonett, E. P. Kvale, M. A. Chan, T. M. Demko, 1996. Science, 273, 100. F. D. Stacey, 1977. Physics of the Earth, second edition. John Willey & Sons. J. W. Wells, 1963. Nature, 197, 948. Caption Figure 1, the evolution of Moon’s distance and the length of day of the earth with the age of the Earth-Moon system. Solid lines are calculated according to the dark matter field fluid model. Data sources: the Moon distances are from Kvale and et al. and for the length of day: (a and b) are from Scrutton ( page 186, fig. 8), c is from Sonett and et al. The dash line marks the current age of the Earth-Moon system. Figure 2, the evolution of Solar days of year with the age of the Earth-Moon system. The solid line is calculated according to dark matter field fluid model. The data are from Wells (3.9 ~ 4.435 billion years range), Sonett (3.6 billion years) and current age (4.5 billion years). Figure 3, the skeptical description of the evolution of Mars’s length of day and the solar days/Mars year with the age of the Mars (assuming that the Mars’s age is about 4.5 billion years). The vertical dash line marks the current age of Mars. Figure 1, Moon's distance and the length of day of Earth change with the age of Earth-Moon system The age of Earth-Moon system (109 years) 0 1 2 3 4 5 Distance Length of day Roche's limit Hansen's result Figure 2, the solar days / year vs. the age of the Earth The age of the Earth (109 years) 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6

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