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 6∈ V (P ) and a 6∈ V (Q), such that c is adjacent to every vertex of V (P ) and a has… CONTINUE READING