Shifting paths to avoidable ones

TL;DR

This paper proves that in any graph, induced paths can be transformed into avoidable induced paths through a sequence of shifts, extending previous results and exploring reconfiguration properties of paths.

Contribution

It strengthens existing theorems by showing all induced paths can be reconfigured into avoidable ones via shifts, broadening understanding of path structures.

Findings

Induced paths can be transformed into avoidable paths through shifts.

The approach applies to not necessarily induced paths and walks.

The result does not extend to trails or isometric paths.

Abstract

An extension of an induced path $P$ in a graph $G$ is an induced path $P'$ such that deleting the endpoints of $P'$ results in $P$. An induced path in a graph is said to be avoidable if each of its extensions is contained in an induced cycle. In 2019, Beisegel, Chudovsky, Gurvich, Milani\v{c}, and Servatius conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced path of the same length, and proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. In the present paper, using a similar approach, we strengthen their result from a reconfiguration point of view. Namely, we show that in every graph, each induced path can be transformed to an avoidable one by a sequence of shifts, where two induced $k$-vertex paths are shifts of each other if their union is an induced path with $k+1$ vertices. We also obtain analogous results for not necessarily induced paths and for walks. In contrast, the statement cannot be extended to trails or to isometric paths.