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1 Characterizing Randomly P k -Decomposable Graphs for k 9 Robert Molina, Alma College Myles McNally, Alma College Ken Smith, Central Michigan University Abstract A graph G is randomly H decomposable if every family of edge disjoint subgraphs of G, each subgraph isomorphic to H, can be extended to an H decomposition of G. Let P k denote a path of length k. In this paper we characterize randomly P k decomposable graphs for k Introduction In this paper we refer to finite simple undirected graphs simply as graphs. We denote the vertex and edge sets of a graph G by V (G) and E(G) respectively. The size of a graph is the cardinality of its edge set. If G and H are graphs, then G + H will denote the graph with vertex set V (G) V (H) and edge set E(G) E(H). If H is a subgraph of G, we define G H to be the graph obtained by first forming the graph with vertex set V (G) and edge set E(G) E(H), and then deleting from this graph any isolated vertices. A path with n edges is denoted P n and a cycle with n edges is denoted C n. Degree one vertices of a path are called end vertices while degree two vertices are called internal vertices. If G and H are graphs, we refer to a subgraph of G that is isomorphic to H as an H-subgraph of G. An H decomposition of G is a family of edge disjoint H subgraphs of G whose union is G, in which case we say G is H decomposable. WesayG is randomly H decomposable if any edge disjoint family of H subgraphs of G can be extended to an H decomposition of G. The set of all randomly H decomposable graphs is denoted RD(H). The set of all connected graphs in RD(H) that are are the union of n edge disjoint H subgraphs is denoted RD(H, n). Note that if G is randomly H decomposable, then we can select any H subgraph of G and remove its edges, then find any remaining H subgraph and remove its edges, and continue in this manner until all of the edges of G have been removed. For example, K 1,6 is randomly P 2 decomposable since removal of any P 2 subgraph yields a K 1,4 and removal of any remaining P 2 subgraph leaves a P 2. The graph P 6, which is P 2 decomposable, is not randomly P 2 decomposable. The concept of random decomposition was introduced by Ruiz in [6]. In that paper he characterizes randomly H decomposable graphs for the 1

2 cases H = K 1,2 and H = 2K 2. Barrientos, Bernasconi, Jeltsch and Tronosco have characterized RD(H) when H is the star K 1,n [2]. Randomly K n decomposable graphs were characterized by Smith and Kabell [7] (the proof also appears in [3]). Beineke, Hamburger and Goddard use the concept of forbidden subgraphs to characterize RD(P k ) for 3 k [3]. They also characterize randomly tk 2 decomposable graphs with sufficiently many edges. Arumugam and Meena characterize randomly H decomposable graphs for H isomorphic to one of the following disconnected graphs: K n K 1,n 1, K n K 1,n, C 4 P 2,2C 3 or 3K 2 [1]. Caro, Rojas and Ruiz give a forbidden subgraph characterization for randomly decomposable graphs in [4], and then use this characterization to show that determining whether a graph is a member of RD(H) can be done in polynomial time. The goal of this paper is to characterize randomly P k decomposable graphs for k 9, extending the results of Beineke et al which went up to k =. These results were obtained using a computer program written in the Java programing environment. The core algorithm is discussed in section 2. Some of the efficiency of the final program is based on several structural results for randomly P k decomposable graphs not reported here. These results will be discussed in a manuscript currently under preparation. The main goal of this paper is to present the graphs that we have found and the basic approach to their generation. The characterizations of RD(P k ), k 9 appear in section 3. Some open questions regarding randomly P k decomposable graphs are presented in section The Algorithm At heart, the generation of randomly P k -decomposable graphs is simple. Take an element of RD(P k,n) and consider all possible ways of adding a P k subgraph to it. For each resulting graph, check to see if it is randomly P k decomposable. In this way obtain all elements of RD(P k,n+ 1). Thus, done in a breadth-first fashion, RD(P k, 1) is used to generate RD(P k, 2), which in turn is used to generate RD(P k, 3), then RD(P k, 4), and so on. The graphs generated by our algorithm have vertex sets consisting of consecutive integers. If G is graph with vertices numbered 0 through m, then a reference string for G is a k string whose terms come from the set {, 0, 1, 2,...,m} such that integer terms in the string can occur at most once. These strings give directions for adding a P k subgraph H to G. An integer a in position i of the reference string means that a will be the i-th vertex of the new path. The terms in the string correspond to vertices of the new path that are not vertices of G. The star vertices of the new path are eventually given consecutive integer values. In figure 1 the reference string is used to add a new P 9 to the graph shown at left in figure 1 (also a P 9 ). In this example the resulting graph is an element of RD(P 9, 2). A reference string such as the one in this example that produces 2

3 a randomly P k -decomposable graph is said to be a productive string for the graph it has extended * * 4 * 2 * 6 * * Figure 1 By running through all reference strings for G we consider all possible ways to add a P k subgraph H to G. In some cases G + H will be a multigraph which does not need to be considered, but otherwise we must test the graph G + H to determine if it is randomly P k -decomposable. The difficulty with this approach is that there are just too many reference strings if G has very many vertices. If we wish to add a P k subgraph to a graph with n vertices, then the number of reference strings that must be considered is k c(k, i) p(n, i). i=0 Clearly this count grows explosively. For instance, in order to add a third path to the graph on 14 vertices shown at right in figure 1, we would need to consider 216,730,118,331 reference strings. Suppose that G RD(P k,n), and we wish to extend G by adding another P k subgraph H. Say that G = H 1 H 2... H n where the H i are the P k subgraphs that have been added by the algorithm. A brute force approach would have us consider all possible reference strings for G. The efficiency of our algorithm is due to the fact that it only considers reference strings that will add a P k subgraph H such that for 1 i n, (G H i )+H RD(P k,n). We call reference strings with this property nice. Note that the set of all nice reference strings for G contains the set of all productive reference strings for G. By considering only nice strings, we dramatically reduce the total number of reference strings that need to be considered. For instance, there are only 296 nice reference strings for the graph shown at right in figure 1. The key to finding nice reference strings for G is keeping track of the work that is done while computing RD(P k,n). In particular, our algorithm 3

4 keeps track of all productive reference strings that were used to generate RD(P k,n) from RD(P k,n 1). This information is then used to obtain for each i the list L i of all productive reference strings for G H i. For every element (s 1, s 2,...,s n ) L 1 L 2... L n the algorithm tries to construct a nice reference string s for G. It does so by the following rules: 1. If the j-th term of each s i is, then choose to be j-th term of s. 2. If v is a vertex of G and if the j-th term of each s i is v, then choose v to be j-th term of s. 3. If the j-th term of exactly one s i,says t,is, and the j-th term of all other s i s is equal to some vertex w such that w V (H t ) but w V (G H t ), then choose w to be j-th term of s. If for some j, none of the conditions for rules 1, 2 or 3 apply, then (s 1, s 2,...s n ) does not yield a nice reference string s for G. Otherwise we claim that the string s generated will be a nice reference string for G. We also claim that all nice strings for G will be obtained in this way, and hence all productive strings for G will be generated. Some comments should be made at this point. Our claim that the algorithm uses information obtained in computing RD(P k,n) to generate RD(P k,n+ 1) is only valid for n 2. Thus the computation of RD(P k, 2) requires that we start with a single P k and consider all reference strings for extending it. We refer to this part of the algorithm as phase I, and to the part where we generate RD(P k,n), n 3, as phase II. Figure 2 gives time requirements for phase I and phase II, programmed in Java running on a Macintosh with 1-gigahertz dual processors under OSX. Time in Minutes Required to Generate RD(Pk,n), n 8 Path Length Number of Graphs Time for Phase I Figure 2 Time for Phase II Total Time

5 Finally, we briefly explain how we can arrive at a characterization for RD(P k ) having only computed a finite number of the sets RD(P k,n). We can do this because, at least for k 9, as n increases the variety of graphs we find in RD(P k,n) decreases. For example, since all graphs in RD(P 2, 3) are isomorphic to K 1,6, and all graphs in RD(P 2, 4) are isomorphic to K 1,8, one can easily argue that all graphs RD(P 2,n) are isomorphic to K 1,2n. The situation is similar for 3 k 9 in the sense that, for large enough n, all elements of RD(P k,n) will fall into one of several easily described families. These families will be described in the next section. 3. Characterizations of RD(P k ),k 9 In this section we characterize randomly P k decomposable graphs for k 9. To do this it is sufficient to describe the connected elements of RD(P k ) since a graph is randomly P k decomposable if and only if its components are. We need some notation in order to describe a particular type of graph that is frequently randomly path decomposable. Let n be a positive integer and for 1 i n let H i be the path H i = v i0 v i1 v i2...v ik. Let S be a nonempty subset of {0, 1, 2,...,k} and let G be the graph obtained from the paths H i by identifying the vertices v 1j, v 2j..., v nj for each j S. Then G is called an n braid. Note that the vertices of the paths which were identified become vertices of degree n or 2n in G. These identification points partition each of the paths into 1 or more subpaths. Suppose that when the path H i is traversed from v i0 to v ik, the lengths of these subpaths, in order, are a 1, a 2,..., a t.if0 Sand k S, then we say that G is an n braid of type (a 1,a 2,...,a t ). Similarly, n braids of type [a 1,a 2,...,a t ), (a 1,a 2,...,a t ] and [a 1,a 2,...,a t ] refer respectively to the cases: 0 S and k S, 0 S and k S and 0 S and k S. Braids of type (3, 2, 3), (2, 2, 2, 2) and [4, 4] are shown in figure 3 below. The reader can verify that each of these examples is randomly P 8 decomposable. Figure 3. Braids Anyone who has thought very much about randomly P k decomposable graphs has probably encountered graphs such as these, as many of them turn out to be randomly path decomposable. For example, 2 braids of type [k] (even cycles) are randomly P k decomposable, and 2 braids of type [k, k] (figure eights) and are randomly P 2k decomposable.

6 The following theorems characterize RD(P k ), for 2 k 9. The characterizations for 2 k can also be found in [3], but are included here for completeness. Theorem 1. A graph G is an element of RD(P 2 ) if and only if the nontrivial components of G are isomorphic to either C 4 or an n braid of type (1,1), i.e., a star with an even number of edges. Theorem 2. A graph G is an element of RD(P 3 ) if and only if the nontrivial components of G are isomorphic to P 3 or one of the graphs shown in figure 4. Theorem 3. A graph G is an element of RD(P 4 ) if and only if the nontrivial components of G are isomorphic to P 4,ann braid of type (2,2), or one of the graphs shown in figure. Theorem 4. A graph G is an element of RD(P ) if and only if the nontrivial components of G are isomorphic to P or one of the graphs shown in figure 6. Figure 4 Figure Figure 6 Theorem. A graph G is an element of RD(P 6 ) if and only if the nontrivial components of G are isomorphic to P 6,ann braid of type (3,3), an n braid of type (2,2,2), or one of the graphs shown in figure 7. Figure 7 Theorem 6. A graph G is an element of RD(P 7 ) if and only if the nontrivial components of G are isomorphic to P 7,ann braid of type (2,2,3), or one of the graphs shown in figure 8. 6

7 Figure 8 Theorem 7. A graph G is an element of RD(P 8 ) if and only if the nontrivial components of G are isomorphic to P 8,ann braid of type (4,4), an n braid of type (3,2,3), an n braid of type (2,2,2,2), or one of the graphs shown in figure 9. Figure 9 The reader may have noticed for k 8, the only connected elements of RD(P k ) not contained in RD(P k, 2) were n braids. This might lead one to conjecture that n braids are the only examples of randomly P k decomposable graphs that are the union of three or more P k s. Examination of randomly P 9 decomposable graphs shows that this conjecture is premature. Theorem 8. A graph G is an element of RD(P 9, 2) if and only if it is isomorphic to one of the graphs shown in figure 10. 7

8 Figure 10 8

9 Theorem 9. A graph G is an element of RD(P 9, 3) if and only if it is isomorphic to one of the graphs shown in figure 11. Figure 11 Theorem 10. A graph G is an element of RD(P 9, 4) if and only if it is isomorphic to one of the graphs shown in figure 12. Figure 12 In order to complete the characterization of RD(P 9 ) we need to describe two more families of graphs. We first define twisted braids. A twisted braid of type I is formed by taking an m braid and an n braid, both of type (3,2,2,2), and twisting them together by identifying vertices in three places as shown at left in figure 13. Here the n braid is shown in bold and the identified vertices are shaded. In a similar way we can twist an m braid and a P 9 together, as shown at right in figure 13, to form a type II twisted 9

10 braid. The reason that a P 9 is used instead of an n braid in the type II twisted braid is that in the latter case multiple edges would result. m n. m. m+n Type I twisted braid m Type II twisted braid Figure 13 The final family of graphs are called branching graphs. Graphs from this family start with an n braid of type (3,2] as a base, with a number of rooted branches attached to the degree n vertex (shaded) as shown in figure 14. The branches are of two types, A and B. Type A branches are a union of two P 4 s and isomorphic to K 2,4, and type B branches are the union of one or more P 4 s that form a braid of type [2,2). Note that the family of branching graphs contains (3,2,4)-braids (take each branch to be a P4) and (3,2,2,2)-braids (take a single type B branch). Any collection of type A and B branches will yield a randomly P 9 -decomposable graph provided that the number of P s in the base equals the number of P 4 s in the branches. Type A and B branches. Type A branch Figure 14.. Type B branch The following theorem completes our characterization of RD(P 9 ). Theorem 11. A graph G is an element of RD(P 9,n) and n, then G is an n braid of type (3,2,4), (3,3,3), or (3,2,2,2), a twisted braid of type I or II, or a branching graph. 10

11 4. Some Open Problems Given the wide variety of randomly path decomposable graphs we have found so far, it seems unlikely that a simple characterization for all such graphs can be found. But the vast majority of the graphs we did find were separable, and 2 connected examples seem to be relatively rare. Problem 1: Is there a simple characterization for 2 connected randomly P k decomposable graphs? There are infinite families of 2 connected randomly P k decomposable graphs: The cycle C 2k is randomly P k decomposable, and a 4 braid of type [k] (4 paths of length k with end vertices identified) is randomly P 2k decomposable. A third such family is discussed in []. A graph from any one of these three families will have all vertex degrees even and be the union of two paths. Problem 2: The graphs K 4, K 2,3 and K 3,4 are examples of 2 connected randomly P k decomposable graphs which have a vertex of odd degree. Do all other 2 connected randomly P k decomposable graphs have all vertex degrees even? Problem 3: Are all of the 2 connected randomly P k decomposable graphs elements of RD(P k, 2)? References [1] S. Arumugam, S. Meena, Graphs that are randomly packable by some common disconnected graphs, Indian J. Pure Appl. Math., 29(11) (1998) [2] C. Barrientos, A. Bernasconi, E. Jeltsch, C. Tronosco, S. Ruiz, Randomly star decomposable graphs, Cong. Num., 64 (1988) [3] L. Beineke, P. Hamburger, W. Goddard, Random Packings of Graphs, Discrete Math., 12 (1994) 4 4 [4] Y. Caro, J. Rojas, S. Ruiz, A forbidden subgraphs characterization and a polynomial algorithm for randomly decomposable graphs, Czechoslovak Math. J., 46 (1996) [] R. Molina, On randomly P k decomposable graphs, Cong. Num., 148 (2001) [6] S. Ruiz, Randomly Decomposable Graphs, Discrete Math., 7 (198) [7] Smith and Kabell, manuscript [8] S. Somasundaram, A. Nagarajan, On randomly packable graphs, Acta Cienc. Indica Math., 23, No. 3 (1997)

AALBORG UNIVERSITY. A short update on equipackable graphs. Preben Dahl Vestergaard. Department of Mathematical Sciences. Aalborg University

AALBORG UNIVERSITY. A short update on equipackable graphs. Preben Dahl Vestergaard. Department of Mathematical Sciences. Aalborg University AALBORG UNIVERSITY A short update on equipackable graphs by Preben Dahl Vestergaard R-2007-06 February 2007 Department of Mathematical Sciences Aalborg University Fredrik Bajers Vej 7 G DK - 9220 Aalborg