The (theta, wheel)-free graphs Part III: cliques, stable sets and coloring

TL;DR

This paper studies a special class of graphs excluding certain subgraphs called thetas and wheels, and develops polynomial algorithms for key problems like maximum weight clique, stable set, and coloring, with bounds on chromatic number.

Contribution

It provides polynomial algorithms for maximum weight clique, stable set, and coloring in (theta, wheel)-free graphs, and establishes chromatic bounds and 3-clique-colorability.

Findings

Polynomial algorithms for maximum weight clique and stable set.

Chromatic number bounded by max{ω,3} for these graphs.

The class is 3-clique-colorable.

Abstract

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three paths between the same pair of distinct vertices so that the union of any two of the paths induces a hole. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins, and consequently obtain a polynomial time recognition algorithm for the class. In this paper we further use this decomposition theorem to obtain polynomial time algorithms for maximum weight clique, maximum weight stable set and coloring problems. We also show that for a graph $G$ in the class, if its maximum clique size is $\omega$, then its chromatic number is bounded by max$\{\omega,3\}$, and that the class is 3-clique-colorable.