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 $ψ_k(G)$. We give estimates -- mostly upper bounds -- on $ψ_k(G)$ 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.