  Given a graph $G=(V,E)$ and a positive integer $k\ge2$, a $k$-path vertex
cover is a subset of vertices $F$ such that every path on $k$ vertices in $G$
contains at least one vertex from $F$. A minimum $k$-path vertex cover in $G$
is a $k$-path vertex cover with minimum cardinality and its cardinality is
called the {\it $k$-path vertex cover number} of $G$. In the {\it $k$-path
vertex cover problem}, it is required to find a minimum $k$-path vertex cover
in a given graph. In this paper, we present a brief survey of the current state
of the art in the study of the $k$-path vertex cover problem and the $k$-path
vertex cover number.
