### Abstract

The bull is the graph consisting of a triangle and two disjoint pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is the first paper in a series of three. The goal of the series is to explicitly describe the structure of all bull-free graphs. In this paper we study the structure of bull-free graphs that contain as induced subgraphs three-edge-paths P and Q, and vertices c∉ V(P) and a∉ V(Q), such that c is adjacent to every vertex of V(P) and a has no neighbor in V(Q). One of the theorems in this paper, namely 1.2, is used in Chudnovsky and Safra (2008) [9] in order to prove that every bull-free graph on n vertices contains either a clique or a stable set of size n14, thus settling the Erdös-Hajnal conjecture (Erdös and Hajnal, 1989) [17] for the bull.

Original language | English (US) |
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Pages (from-to) | 233-251 |

Number of pages | 19 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 102 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

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

### Keywords

- Bull-free graphs
- Decomposition theorems
- Graph structure
- Induced subgraphs