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arXiv · 2203.11667

TS-Reconfiguration of $k$-Path Vertex Covers in Caterpillars for $k \geq 4$

Duc A. Hoang
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Abstract

A $k$-path vertex cover ($k$-PVC) of a graph $G$ is a vertex subset $I$ such that each path on $k$ vertices in $G$ contains at least one member of $I$. Imagine that a token is placed on each vertex of a $k$-PVC. Given two $k$-PVCs $I, J$ of a graph $G$, the $k$-Path Vertex Cover Reconfiguration ($k$-PVCR) under Token Sliding ($\mathsf{TS}$) problem asks if there is a sequence of $k$-PVCs between $I$ and $J$ where each intermediate member is obtained from its predecessor by sliding a token from some vertex to one of its unoccupied neighbors. This problem is known to be $\mathtt{PSPACE}$-complete even for planar graphs of maximum degree $3$ and bounded treewidth and can be solved in polynomial time for paths and cycles. Its complexity for trees remains unknown. In this paper, as a first step toward answering this question, for $k \geq 4$, we present a polynomial-time algorithm that solves $k$-PVCR under $\mathsf{TS}$ for caterpillars (i.e., trees formed by attaching leaves to a path).

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Duc A. Hoang. 2022-08-08. TS-Reconfiguration of $k$-Path Vertex Covers in Caterpillars for $k \geq 4$. https://doi.org/10.20429/tag.2023.10108

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