(Even hole, triangle)-free graphs revisited

2026-04-02

AI Summary

This study investigates the structural and algorithmic properties of (even hole, triangle)-free graphs. By introducing the broader class of (theta, triangle, wac)-free graphs, the authors establish a stronger structural theorem that precisely characterizes its basic building blocks and separation structures. They prove that the treewidth of this graph class is at most 4, improving the previously known upper bound of 5. Leveraging this refined structural understanding, they derive a concise criterion for planarity and design an efficient recognition algorithm with time complexity $O(|V|^4|E|)$. The work advances the field through three key contributions: a sharper structural characterization, a tight bound on treewidth, and an improved algorithmic framework.

Abstract

We revisit a classical paper about (even hole, triangle)-free graphs [Conforti, Cornuéjols, Kapoor and Vu\v sković, 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.

Problem

• even hole
• triangle-free
• graph structure
• treewidth
• theta-free

Innovation

• even hole-free graphs
• triangle-free graphs
• treewidth
• structure theorem
• recognition algorithm