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 n 1 4 , thus settling the Erdös–Hajnal conjecture (Erdös and Hajnal, 1989) [17] for the bull.
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