  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).
