### 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 X_{1}, . . 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 H_{1} and H_{2}, 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 ({H_{1}, H_{2}, J_{1}, J_{2}}, 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 H_{1}, H_{2}, we say that G is {H_{1}, H_{2}}-split if V(G)=X_{1}∪X_{2}, 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.

Original language | English (US) |
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Pages (from-to) | 15-29 |

Number of pages | 15 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 106 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2014 |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Keywords

- Cograph
- Induced subgraphs
- Ramsey theorem

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*106*(1), 15-29. https://doi.org/10.1016/j.jctb.2014.01.001