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

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 (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 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 ≥ 4, we present a polynomial-time algorithm that solves k-PVCR under TS for caterpillars (i.e., trees formed by attaching leaves to a path).