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, . . X_P, so that for every i∈{1, . . ., P}, either |X_i|=1, or the subgraph of G induced by X_i 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 J1c and J2c, 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 X_i is {H_i}-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, 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 {H, J}-free graph is {H~,J~}-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.