The (theta, wheel)-free graphs Part IV: induced paths and cycles

TL;DR

This paper studies a special class of graphs excluding certain subgraphs called thetas and wheels, and uses a decomposition theorem to solve problems related to induced paths and cycles within these graphs.

Contribution

It applies a decomposition theorem to address problems of finding induced paths and cycles in theta- and wheel-free graphs, advancing understanding of their structure.

Findings

Decomposition theorem for theta- and wheel-free graphs established

Algorithms for detecting induced paths and cycles developed

Structural properties of these graphs elucidated

Abstract

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. 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. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class.