Avoidability beyond paths

TL;DR

This work generalizes the notion of avoidability from paths to arbitrary two-rooted graphs by introducing the framework of copies, extensions, closability, and the core concepts of inherent and PE-inherent graphs. It develops the pendant extension (PE) method to analyze how copies can be extended and to define limit graphs, proving that all proper PE-sequences yield a unique limit graph for connected underlying graphs. The authors establish both positive results, such as the inherentness of endpoint-rooted paths and several comb-type families, and negative results, via explicit confining graphs (including circulants and cages) demonstrating non-inherence in many cases. The paper thus advances the structural understanding of avoidability across graph families, connects to recent related work, and outlines open questions and generalizations to k-rooted notions (amoebas) with potential implications for broader combinatorial problems.

Abstract

The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milanič, 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].

Figures (29)

• Figure 1: Two-rooted paths for Theorem \ref{['th:not-inher']}
• Figure 2: An infinite PE-sequence of $(P_1,s,t)$, where $V(P_1) = \{s\} = \{t\}$
• Figure 3: A finite PE-sequence of $(C_3,s,t)$, $s\ne t$
• Figure 4: Two PE-sequences for $(C_4,s,t)$, where $s$ and $t$ are adjacent
• Figure 5: Two PE-sequences of $(P_1,s,t)$, where $V(P_1) = \{s\} = \{t\}$

Theorems & Definitions (89)

• Theorem 1.1
• Conjecture 1.2
• Conjecture 1.2: reformulated
• Definition 1.3
• Remark 1.4
• Definition 1.5
• Definition 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 1.9