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Excluding Pairs of Graphs Excluding pairs of graphs Maria Chudnovsky1 Columbia University, New York, NY 10027, USA Alex Scott Mathematical Institute, University of Oxford, 24-29 St Giles', Oxford OX1 3LB, UK Paul Seymour2 Princeton University, Princeton, NJ 08544, USA September 3, 2012; revised September 11, 2013 1Supported by NSF grants DMS-1001091 and IIS-1117631. 2Supported by ONR grant N00014-10-1-0680 and NSF grant DMS-0901075. Abstract For a graph G and a set of graphs H, we say that G is H-free if no induced subgraph of G is isomorphic to a member of H. Given an integer P > 0, a graph G, and a set of graphs F, we say that G admits an (F;P )-partition if the vertex set of G can be partitioned into P subsets X1;:::;XP , so that for every i 2 f1;:::;P g, either jXij = 1, or the subgraph of G induced by Xi is fF g-free for some F 2 F. Our first result is the following. For every pair (H; J) of graphs such that H is the disjoint union c c c of two graphs H1 and H2, and the complement J of J is the disjoint union of two graphs J1 and J2, there exists an integer P > 0 such that every fH; Jg-free graph has an (fH1;H2;J1;J2g;P )-partition. A similar result holds for tournaments, and this yields a short proof of one of the results of [1]. A cograph is a graph obtained from single vertices by repeatedly taking disjoint unions and disjoint unions in the complement. For every cograph there is a parameter measuring its complexity, called its height. Given a graph G and a pair of graphs H1;H2, we say that G is fH1;H2g-split if V (G) = X1 [ X2, where the subgraph of G induced by Xi is fHig-free for i = 1; 2. Our second result is that for every integer k > 0 and pair fH; Jg of cographs each of height k + 1, where neither of H; J c is connected, there exists a pair of cographs (H;~ J~), each of height k, where neither of H~ c; J~ is connected, such that every fH; Jg-free graph is fH;~ J~g-split. Our final result is a construction showing that if fH; Jg are graphs each with at least one edge, then for every pair of integers r; k there exists a graph G such that every r-vertex induced subgraph of G is fH; Jg-split, but G does not admit an (fH; Jg; k)-partition. 1 Introduction All graphs in this paper are finite and simple. Let G be a graph. For X ⊆ V (G), we denote by GjX the subgraph of G induced by X. The complement of G, denoted by Gc, is the graph with vertex set V (G) such that two vertices are adjacent in G if and only if they are non-adjacent in Gc.A clique in G is a set of vertices all pairwise adjacent; and a stable set in G is a set of vertices all pairwise non-adjacent. For disjoint X; Y ⊆ V (G), we say that X is complete (anticomplete) to Y if every vertex of X is adjacent (non-adjacent) to every vertex of Y . If jXj = 1, say X = fxg, we say \x is complete (anticomplete) to Y " instead of \fxg is complete (anticomplete) to Y ". We denote by Kn the complete graph on n vertices, and by Sn the complement of Kn. A graph is complete multipartite if its vertex set can be partitioned into stable sets, all pairwise complete to each other. For graphs H and G we say that G contains H if some induced subgraph of G is isomorphic to H. Let F be a set of graphs. We say that G is F-free if G contains no member of F. If jFj = 1, say F = fF g, we write \G is F -free" instead of \G is fF g-free". For a pair of graphs fH1;H2g, we say that G is fH1;H2g-split if V (G) = X1 [ X2, and GjXi is Hi-free for i = 1; 2. We remind the reader that a split graph is a graph whose vertex set can be partitioned into a clique and a stable set; thus in our language split graphs are precisely the graphs that are fK2;S2g-split. Ramsey's theorem can be restated in the following way: for every pair of integers m; n > 0 there exists an integer P , such that for every fSm;Kng-free graph G, V (G) can be partitioned into at most P well-understood parts (in fact, each part is a single vertex). One might ask whether a similar statement holds for more general pairs of graphs than just fSm;Kng (adjusting the definition of \well-understood"). c For instance, a result of [2] implies that if G is fC4;C4g-free (where C4 is a cycle on four vertices), then V (G) can be partitioned into three parts, each of which induces either a complete graph, or a graph with no edges, or a cycle of length five. In [3] two of us made progress on this question, but to state the result we first need a definition. Given an integer P > 0, we say that a graph G admits an (F;P )-partition if V (X) = X1 [ ::: [ XP such that for every i 2 f1;:::;P g, either jXij = 1 or GjXi is fF g-free for some F 2 F. Please note that the first alternative in the definition of an (F;P )-partition (the condition that jXij = 1) is only necessary when no graph in F has more than one vertex. We proved the following: 1.1. For every pair of graphs (H; J) such that Hc and J are complete multipartite, there exist integers k; P > 0 such that every fH; Jg-free graph admits a (fKk;Skg;P )-partition. and its immediate corollary: 1.2. For every pair of graphs (H; J) such that Hc and J are complete multipartite, there exists an integer k > 0 such that every fH; Jg-free graph is fKk;Skg-split. The first goal of this paper is to generalize 1.1 further. Let H be a graph. A component of H is a maximal connected subgraph of H. A graph is anticonnected if its complement is connected. An anticomponent of H is a maximal anticonnected induced subgraph of H. We denote by c(H) the set of components of H, and by ac(H) the set of anticomponents of H. We remark that for every non-null graph G, at least one of c(G) or ac(G) equals fGg. We prove the following generalization of 1.1 (please note that 1.3 is trivial whenever H is connected or J is anticonnected): 1.3. For every pair of graphs (H; J) there exists an integer P such that every fH; Jg-free graph admits a (c(H) [ ac(J);P )-partition. 1 Please note that applying 1.3 with J a complete graph and H a graph with no edges gives Ramsey's theorem. Using ideas similar to those of our proof of 1.3, we also give a short proof of one of the results of [1]. An anonymous referee asked whether the conclusion of 1.3 can be strengthened to say that 0 0 V (G) = X1 [ ::: [ XP where for each i 2 f1;:::;P g either jXij = 1 or GjXi is (H ;J )-free for some H0 2 c(H) and J 0 2 ac(J). The answer to this question is \no", because of the following example: c let n be a positive integer, G be the star K1;n, J be a cycle of length four, and H = J . Then G is fH; Jg-free, and the proposed strengthening would say that V j(G)j ≤ P for some fixed integer P , which is false, since n can be made arbitrarily large. Next let us generalize the notion of a complete multipartite graph. A cograph is a graph obtained from 1-vertex graphs by repeatedly taking disjoint unions and disjoint unions in the complement. In particular, G is either not connected or not anticonnected for every cograph G with at least two vertices, and therefore for every cograph G with at least two vertices, exactly one of G; Gc is connected. It follows from [4] that cographs are precisely the graphs that have no induced three- edge paths. We recursively define a parameter, called the height of a cograph, that measures its complexity, as follows. The height of a one vertex cograph is zero. If G is a cograph that is not connected, let m be the maximum height of a component of G; then the height of G is m + 1. If G is a cograph that is not anticonnected, let m be the maximum height of an anticomponent of G; then the height of G is m + 1. We denote the height of G by h(G). We use 1.3 to prove the following: 1.4. Let k > 0 be an integer, and let H and J be cographs, each of height k + 1, such that H is anticonnected, and J is connected. Then there exist cographs H~ and J~, each of height k, such that H~ is connected, and J~ is anticonnected, and every fH; Jg-free graph is fH;~ J~g-split.
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