On the structure of (dart, odd hole)-free graphs

Abstract

A hole is a chordless cycle with at least four vertices. A hole is odd if it has an odd number of vertices. A dart is a graph which vertices $a, b, c, d, e$ and edges $ab, bc, bd, be, cd, de$. Dart-free graphs have been actively studied in the literature. We prove that a (dart, odd hole)-free graph is perfect, or does not contain a stable set on three vertices, or is the join or co-join of two smaller graphs. Using this structure result, we design a polynomial-time algorithm for finding an optimal colouring of (dart, odd hole)-free graphs. A graph $G$ is perfectly divisible if every induced subgraph $H$ of $G$ contains a set $X$ of vertices such that $X$ meets all largest cliques of $H$, and $X$ induces a perfect graph. The chromatic number of a perfectly divisible graph $G$ is bounded by $ω^2$ where $ω$ denotes the number of vertices in a largest clique of $G$. We prove that (dart, odd hole)-free graphs are perfectly divisible.