Avoidability beyond paths

TL;DR

This paper explores the concept of avoidability in graphs beyond paths, providing new definitions and results for avoidable structures with two terminals, and discusses their implications and limitations.

Contribution

It generalizes the notion of avoidability to arbitrary graphs with two terminals and presents both positive and negative results, expanding the theoretical framework.

Findings

Certain avoidable structures always exist in graphs with specific properties.

Some avoidability results do not extend to all graph classes, indicating limitations.

Connections to recent work suggest broader implications for graph theory.

Abstract

The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milani\v{c}, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order $k$ also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which appear to be related to the recent work by Chudnovsky, Norin, Seymour, and Turcotte [arXiv:2301.13175].