Excluding pairs of graphs

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 ∈ {1,..., P}, either |Xi | = 1, or the subgraph of G induced by Xi is {F}-free for some F ∈ F. Our first result is the following. For every pair (H,J) of graphs such that H is the disjoint union of two graphs H1 and H2, and the complement J c of J is the disjoint union of two graphs Jc1 and J c 2, there exists an integer P> 0 such that every {H,J}-free graph has an ({H1, H2, J1, J2}, 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 {H1, H2}-split if V (G) = X1∪X2, where the subgraph of G induced by Xi is {Hi}-free for i = 1, 2. Our second result is that for every integer k> 0 and pair {H,J} of cographs each of height k + 1, where neither of H,Jc is connected, there exists a pair of cographs (H̃, J̃), each of height k, where neither of H̃c, J ̃ is connected, such that every {H,J}-free graph is {H̃, J̃}-split. Our final result is a construction showing that if {H,J} 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 {H,J}-split, but G does not admit an ({H,J}, k)-partition