  For an integer $k\ge 3$, a $k$-path vertex cover of a graph $G=(V,E)$ is a
set $T\subseteq V$ that shares a vertex with every path subgraph of order $k$
in $G$. The minimum cardinality of a $k$-path vertex cover is denoted by
$\psi_k(G)$. We give estimates -- mostly upper bounds -- on $\psi_k(G)$ in
terms of various parameters, including vertex degrees and the number of
vertices and edges. The problem is also considered on chordal graphs and planar
graphs.
