Problem
Research questions and friction points this paper is trying to address.
Generalize avoidable paths to other graph structures
Define avoidability for graphs with terminal vertices
Provide positive and negative results on avoidability
Avoidability beyond paths
🤖 AI Summary
This work investigates the generalization of the “avoiding path” concept—from simple graphs to arbitrary subgraphs with two designated terminals—and systematically establishes a formal definition of generalized avoidability. Methodologically, it employs combinatorial graph theory, structural induction, minimal counterexample analysis, and integrates Chudnovsky et al.’s (2023) recent connectivity decomposition techniques. The paper delivers the first necessary and sufficient condition for the extendability of avoidability, thereby delineating its theoretical boundaries: it proves robust extendability across multiple graph classes, while refuting several natural conjectures to precisely characterize feasibility limits. The core contribution is a unified framework for avoidability applicable to any two-terminal subgraph, yielding tight characterizations—both positive (sufficient conditions) and negative (inherent limitations)—that jointly establish the precise scope of generalization.
📝 Abstract
The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milaniv{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].
Innovation
Methods, ideas, or system contributions that make the work stand out.
Generalizes avoidability to graphs with terminals
Provides positive and negative avoidability results
Links findings to recent graph theory research
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