Abstract. 1. Introduction

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1 MATCHINGS IN 3-DOMINATION-CRITICAL GRAPHS: A SURVEY by Nawarat Ananchuen * Department of Mathematics Silpaorn University Naorn Pathom, Thailand nawarat@su.ac.th Abstract A subset of vertices D of a graph G is a dominating set for G if every vertex of G not in D is adjacent to one in D. The cardinality of any smallest dominating set in G is denoted by γ(g) and called the domination number of G. Graph G is said to be γ-edgecritical if γ(g + e) < γ(g) for each edge e / E(G) and is said to be γ-vertex-critical if γ(g v) < γ(g), for every vertex v in G. In this paper, we explore matching properties in both 3-edge-critical graphs and 3-vertex-critical graphs. 1. Introduction Let G denote a finite undirected graph with vertex set V (G) and edge set E(G). A perfect matching in a graph G is a matching which covers all the vertices of G and a near-perfect matching is one which covers all but exactly one of the vertices of G. A graph G is factor-critical if G {v} has a perfect matching for every vertex v V (G) and is bicritical if G {u, v} has a perfect matching for every pair of distinct vertices u, v V (G). More generally, a graph G is said to be -factor-critical if G S has a perfect matching for every set of vertices in G. A set S V (G) is a (vertex) dominating set for G if every vertex of G either belongs to S or is adjacent to a vertex of S. The minimum cardinality of a vertex dominating set in graph G is called the (vertex) domination number (or simply the domination number) of G and is denoted by γ(g). Graph G is said to be γ-edge-critical if γ(g + e) < γ(g) for each edge e / E(G). Clearly, then γ(g + e) = γ(g) 1 for every edge e / E(G). The structure of γ-edge-critical graphs remains far from completely understood when γ 3. Sumner and Blitch [SB] were the first to study 3-edge-critical graphs. They gave a characterization of -edge-critical graphs for = 1 and 2. They showed that the only 1-edge-critical graphs are the complete graphs {K n n 1} and a graph G is 2-edgecritical if and only if it is the complement of a union of stars; i.e., Ḡ = t i=1 K 1, n i. for t 1. They also showed that a disconnected 3-edge-critical graph consists of the disjoint union of a 2-edge-critical graph and a complete graph. For summaries of most nown results, see [HHS;Chapter 16] as well as [FTWZ]. The related, yet different, concept of vertex criticality with respect to domination * wor supported by the Thailand Research Fund Grant #BRG

2 number has also received attention. Graph G is said to be γ-vertex-critical if γ(g v) < γ(g), for every vertex v in G. Clearly, then γ(g v) = γ(g) 1, for every vertex v in G. The structure of such graphs remains relatively unexplored, even in the case γ = 3. The study of γ-vertex-critical graphs was begun by Brigham, Chinn and Dutton [BCD] and continued by Fulman, Hanson and MacGillivray [FHM]. Clearly, the only 1-vertex-critical graph is K 1 (a single vertex). Brigham, Chinn and Dutton [BCD] pointed out that the 2-vertex-critical graphs are precisely the family of graphs obtained from the even complete graphs K 2n by deleting a perfect matching. For γ > 2, however, an understanding of the structure of γ-vertex-critical graphs is far from complete. In this paper, we present results on matchings in both 3-edge-critical graphs and 3- vertex-critical graphs. Section 2 contains results on matching properties in 3-edge-critical graphs. Matching results for 3-vertex-critical graphs comprise Section 4. Finally, we refer the reader to [LP] for further notation, terminology and bacground for matching theory edge-criticality and -factor-criticality We begin with a result concerning the existence of a perfect matching, proved by Sumner and Blitch [SB], and a near-perfect matching proved by Ananchuen and Plummer [AP1]. Theorem 2.1. Let G be a connected 3-edge-critical graph. (i) Then if V (G) is even, G contains a perfect matching, while (ii) if V (G) is odd, G contains a near-perfect matching. The following results concerning -factor-criticality for = 1, 2 were proved by Ananchuen and Plummer [AP1]. Theorem 2.2. G is factor-critical. Let G be a 2-connected 3-edge-critical graph having odd order. Then Theorem 2.3. If G is a 3-connected 3-edge-critical graph with minimum degree at least 4 and having even order, then G is bicritical. They also showed that the lower bound on the connectivity stated in Theorem 2.2 and the minimum degree bound stated in Theorem 2.3 are best possible. However, the minimum degree bound can be relaxed if the graphs are planar. Theorem 2.4. (Ananchuen and Plummer [AP1]) If G is a 3-connected 3-edgecritical planar graph having even order, then G is bicritical. A graph is said to be claw-free if it contains no induced subgraph isomorphic to K 1,3. Plummer [P] proved that if G is a 3-connected claw-free graph of even order, then G is bicritical. If the even graphs involved are 3-edge-critical, the demand on connectivity can be lowered and still guarantee bicriticality. 2

3 Theorem 2.5. (Ananchuen and Plummer [AP1]) Let G be a 2-connected 3-edgecritical claw-free graph of even order. Then if mindeg G 3, G is bicritical. We next present results involving 3-factor-critical graphs proved by Ananchuen and Plummer [AP2]. Theorem 2.6 If G is a 4-connected 3-edge-critical graph of odd order and having minimum degree at least 5, then G is 3-factor-critical. Theorem 2.7 Let G be a 3-connected 3-edge-critical claw-free graph of odd order. Then if mindeg G 4, G is 3-factor-critical. We conclude this section by presenting two conjectures involving matching in 3-edgecritical graphs stated in Ananchuen and Plummer [AP6]. Conjecture 1: Suppose G is a graph with 2 and suppose 1 and V (G) have the same parity. Then if G is -connected and 3-edge-critical with mindeg G + 1, then G is ( 1)-factor-critical. Conjecture 2: Suppose G is a graph with 2 and suppose and V (G) have the same parity. Then if G is -connected and 3-edge-critical with mindeg G + 1 and G is claw-free, then G is -factor-critical. Conjecture 1 is nown to be true when = 2 (Theorem 2., when = 3 (Theorem 2.3) and when = 4 ( Theorem 2.6). Conjecture 2 is nown to be true when = 2 (Theorem 2.5) and when = 3 (Theorem 2.7). However, the proofs of Conjecture 1 when = 4 and Conjecture 2 when = 3 are quite long and difficult. This leads us to thin that settling either of these conjectures for any further values of will be very difficult, if not impossible, using the methods employed for the small values of mentioned above. In our opinion, some new methods must be discovered and utilized vertex-criticality and -factor-criticality In this section, we turn our attention to 3-vertex-critical graphs. We begin with a construction, shown in Ananchuen and Plummer [AP3], which yields an infinite family of new 3-vertex-critical graphs. Let be any positive integer with 5. We proceed to outline the construction of a graph which we will call H,(. The vertex set will consist of two disjoint subsets of vertices called central and peripheral, respectively. Let {v 1,..., v } denote the set of central vertices. The subgraph induced by these central vertices will be the complete graph K with the Hamiltonian cycle v 1 v 2 v v 1 deleted. The peripheral vertices will be ( in number and will be denoted by the symbol {i, j} where the (unordered) pair {i, j} (i j) ranges over all the ( subsets of size 2 of the set 1,...,, except those having j = i + 2 where i + 2 is read modulo. The neighbor set of peripheral vertex {i, j} will be precisely the set of all central vertices, except i and j. There are no edges joining pairs of peripheral vertices. 3

4 Figure 3.1 shows as an example the graph H 6,9. ~{3,4} ~{1,4} ~{4,5} 1 6 ~{2,3} ~{5,6} ~{2,5} ~{3,6} ~{1,2} ~{1,6} Figure 3.1 Note that the graph H,( is ( -connected. Further, each graph H,( can, in turn, be used to create a large number of additional 3-vertex-critical graphs as follows. Partition the set of peripheral vertices into r 3 subsets P 1, P 2, P 3,..., P r and add e i edges to P i for each i = 1,..., r. Here e i can be any integer such that 0 e i ( P i ) 2. The resulting graph is 3-vertex-critical. We denote this graph by H,( ( P 1, P 2,..., P r ) if each P i is complete. It was shown by Sumner and Blitch [SB] that any connected 3-edge-critical graph of even order must contain a perfect matching. In contrast to their result, a connected 3-vertex-critical graph of even order need not contain a perfect matching. The graphs H,( for 6 (which have an even number of vertices when ( is even) are a counterexamples. On the other hand, Ananchuen and Plummer [AP3] established a sufficient condition for 3-vertex-critical graphs to contain a perfect matching as follows. Theorem 3.1. matching. If G is K 1,5 -free 3-vertex-critical of even order, then G has a perfect Note that for infinitely many values of 8 such that ( is even, the graph H,( (1, 1, ( satisfies the hypotheses of Theorem 3.1. For 3-vertex-critical graphs of odd order, Ananchuen and Plummer [AP4] showed that: 4

5 Theorem 3.2. Suppose G is a K 1,5 -free 3-vertex-critical graph of odd order at least 11 with mindeg G > 0. Then G contains a near-perfect matching. Theorem 3.3. If G is a K 1,4 -free 3-vertex-critical graph of odd order with minimum degree at least 3, then G is factor-critical. They also showed that the lower bound on the number of vertices stated in Theorem 3.2 cannot be lowered. Similarly, the hypothesis that the graph be K 1,4 -free in Theorem 3.3 cannot be weaened to say that the graph be K 1,5 -free. The graphs in Figure 3.2 (with r, s 3) are K 1,5 -free 3-vertex-critical connected graphs of odd order with minimum degree at least 3, (in fact, with minimum degree at least 4), but are not factor-critical.... v... K - a perfect matching K - a perfect matching 2r 2s Figure 3.2 Note that if G denotes a graph of the type shown in Figure 3.2, then G v has no perfect matching. Further, G contains K 1,4 as a subgraph. If we increase the connectivity of the graphs involved, however, we believe that one can relax the property of K 1,4 -free to K 1,5 -free. So we set forth the following conjecture. Conjecture 3: If G is a K 1,5 -free 3-vertex-critical 2-connected graph of odd order with minimum degree at least 3, then G is factor-critical. We conclude our paper by presenting an infinite family of claw-free 3-vertex-critical graphs and some matching results for these graphs established by Ananchuen and Plummer [AP5]. For positive integers t, r and s, we construct the graph G(t, r, s) as follows. Let X = {x 1, x 2,..., x t }, Y = {y 1, y 2,..., y r }, T = {u 1, u 2,..., u t, v 1, v 2,..., v r } and S = {z 1, z 2,..., z s, w 1, w 2,..., w s }. Then set V (G(t, r, s)) = X Y T S {a}, thus yielding a set of 2t + 2r + 2s + 1 distinct vertices. Join vertex a to each vertex of S. Form complete graphs on each of X, Y and T and form a complete graph on S, except for the perfect matching {z i w i 1 i s}. Finally, join each x i to each vertex of (T {u i }) {z 1, z 2,..., z s } and join each y i to each vertex of (T {v i }) {w 1, w 2,..., w s }. It is not difficult to show that G(t, r, s) is a claw-free 3-vertex-critical graph. Figure 3.3 shows the graphs G(1, 2, 1) and G(1, 2,. Note that these graphs are 2-connected and 3-connected, respectively. Theorems 3.4, 3.5 and 3.6 guarantee certain connectivity for claw-free 3-vertex-critical graphs, given sufficient minimum degree. The graphs G(1, 2, 1) and G(1, 2, show these assumptions on minimum degree to be best possible. 5

6 G(1, 2, 1) G(1, 2, Figure 3.3 Theorem connected. Let G be a connected claw-free 3-vertex-critical graph. Then G is Theorem 3.5. Let G be a connected claw-free 3-vertex-critical graph. Then if G is of even order or if mindeg G 3, then G is 3-connected. Theorem 3.6 Let G be a connected claw-free 3-vertex-critical graph. Then if mindeg G 5, G is 4-connected. Favaron et al.[ffr] and Liu and Yu [LY] proved independently that if G is ( + 1)- connected, claw-free and of order n, and if n is even, then G is -factor-critical. We then have the following corollary the proof of which is immediate by this fact together with Theorems 3.4, 3.5 and 3.6. Corollary 3.7. (a) Let G be a connected claw-free 3-vertex-critical graph of odd order. Then G is factor-critical. (b) Let G be a connected claw-free 3-vertex-critical graph of even order. Then G is bicritical. (c) Let G be a connected claw-free 3-vertex-critical graph of odd order. mindeg G 5, G is 3-factor-critical. Then if References [AP1] [AP2] N. Ananchuen and M.D. Plummer, Matching properties in domination critical graphs, Discrete Math., 277, 2004, N. Ananchuen and M.D. Plummer, 3-factor-criticality in domination critical graphs, (submitted). 6

7 [AP3] [AP4] [AP5] [AP6] [BCD] [FFR] [FHM] N. Ananchuen and M.D. Plummer, Matchings in 3-vertex-critical graphs: the even case, Networs, 2005, (to appear). N. Ananchuen and M.D. Plummer, Matchings in 3-vertex-critical graphs: the odd case, (submitted). N. Ananchuen and M.D. Plummer, On the connectivity and matchings in 3-vertexcritical claw-free graphs, Utilitas Math., 2005, (to appear). N. Ananchuen and M.D. Plummer, Two conjectures on matching in 3-dominationcritical graphs (submitted). R. Brigham, P. Chinn and R. Dutton, Vertex domination-critical graphs, Networs 18 (1988), O. Favaron, E. Flandrin and Z. Ryjáče, Factor-criticality and matching extension in DCT-graphs, Discuss. Math. Graph Theory 17 (1997), J. Fulman, D. Hanson and G. MacGillivray, Vertex domination-critical graphs, Networs 25 (1995), [FTWZ] E. Flandrin, F. Tian, B. Wei and L. Zhang, Some properties of 3-dominationcritical graphs, Discrete Math., 205, 1999, [HHS] [LP] [LY] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs; Advanced Topics, Marcel Deer, New Yor, L. Lovász and M.D. Plummer, Matching Theory, Ann. Discrete Math. 29, North- Holland, Amsterdam, G. Liu and Q. Yu, On n-edge-deletable and n-critical graphs, Bull. Inst. Combin. Appl. 24 (1998), [P] M.D. Plummer, Extending matchings in claw-free graphs, Discrete Math., 125, 1994, [SB] D.P. Sumner and P. Blitch, Domination critical graphs, J. Combin. Theory Series B, 34, 1983,