The structure of bull-free graphs II and III - A summary

The bull is a graph consisting of a triangle and two pendant edges. A graph is called bull-free if no induced subgraph of it is a bull. This is a summary of the last two papers [2, 3] in a series [1, 2, 3]. The goal of the series is to give a complete description of all bull-free graphs. We call a bull-free graph elementary if it does not contain an induced three-edge-path P such that some vertex c 6∈ V (P ) is complete to V (P ), and some vertex a 6∈ V (P ) is anticomplete to V (P ). Here we prove that every elementary graph either belongs to one a few basic classes, or admits a certain decomposition, and then use this result together with the results of [1] to give an explicit description of the structure of all bull-free graphs.