(Even hole, triangle)-free graphs revisited

TL;DR

This paper revisits and generalizes the structure of (even hole, triangle)-free graphs, providing new theorems, algorithms, and bounds on properties like treewidth and planarity.

Contribution

It introduces a stronger structure theorem for a broader class of graphs and improves bounds on treewidth and recognition algorithms.

Findings

Treewidth of the class is at most 4, improving previous bounds.

Recognition algorithm runs in O(|V|^4|E|) time.

Provides a criterion to determine planarity of graphs in the class.

Abstract

We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornu\'ejols, Kapoor and Vu\v skovi\'c, Triangle-free graphs that are signable without even holes, Journal of Graph Theory, 34(3), 204--220, 2000]. In fact, the previous study describes a more general class, the so called triangle-free odd signable graphs, and we further generalise the class to the (theta, triangle, wac)-free graphs (not worth defining in an abstract). We exhibit a stronger structure theorem, by precisely describing basic classes and separators. We prove that the separators preserve the treewidth and several properties. Some consequences are a recognition algorithm with running time $O(|V(G)|^4|E(G)|)$, a proof that the treewidth of graphs in the class is at most~4 (improving a previous bound of~5), and a simple criterion to decide if a graph in the class is planar.