A Polynomial-Time Algorithm for $k$-Path Vertex Cover Reconfiguration on Trees under Token Sliding

A $k$-path vertex cover ($k$-PVC) of a graph is a set of vertices meeting every path on $k$ vertices. In Token Sliding, one token occupies each vertex of a cover; a move slides one token along an edge to an unoccupied vertex, preserving the cover property. We study whether two covers of a tree are joined by such moves. The case $k=2$ admits a linear-time decision algorithm. For every integer $k\ge 2$, we give an $O(k{n}^2)$-time decision algorithm on trees of order $n$, with $k$ part of the input. Two covers are reachable precisely when their rigid tokens, which never move, occupy the same set and every component remaining after their deletion has the same token count. The proof transforms configurations of equal size without rigid tokens into a canonical configuration built on properly rooted subtrees, whose paths on $k$ vertices pass through their roots. A rooted-subtree recurrence identifies movable candidate tokens. We also give a direct recursive construction between the supplied endpoints, returning at most $8n^2{3}^{\lceil n\mathrm{/}2\rceil }$ slides in $O(n^{10}{3}^{n\mathrm{/}2})$ time. ChatGPT 6 Pro and Claude Fable 5.1 were used to assist with developing and reviewing mathematical arguments and drafting and revising the paper. The author takes full responsibility for the correctness and content of the paper.