On the structure of (claw,bull)-free graphs

TL;DR

This paper characterizes the structure of (claw, bull)-free graphs, showing they are either expansions of paths or cycles or complements of triangle-free graphs, revealing new insights into their composition.

Contribution

It provides a complete structural description of (claw, bull)-free graphs, including their relation to expansions and complements of simpler graph classes.

Findings

Connected (claw, bull)-free graphs are expansions of paths or cycles

They are also complements of triangle-free graphs

The results offer new insights into the structure of triangle-free graphs

Abstract

In this research, we determine the structure of (claw, bull)-free graphs. We show that every connected (claw, bull)-free graph is either an expansion of a path, an expansion of a cycle, or the complement of a triangle-free graph; where an expansion of a graph $G$ is obtained by replacing its vertices with disjoint cliques and adding all edges between cliques corresponding to adjacent vertices of $G$. This result also reveals facts about the structure of triangle-free graphs, which might be of independent interest.