  The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that
the matching number equals the vertex cover number for every induced subgraph
of some graph if and only if that graph is bipartite. Inspired by this result,
we consider the set ${\cal G}_k$ of all graphs such that, for every induced
subgraph, the maximum number of disjoint paths of order $k$ equals the minimum
order of a set of vertices intersecting all paths of order $k$. For $k\in \{
3,4\}$, we give complete structural descriptions of the graphs in ${\cal G}_k$.
Furthermore, for odd $k$, we give a complete structural description of the
graphs in ${\cal G}_k$ that contain no cycle of order less than $k$. For these
graph classes, our results yield efficient recognition algorithms as well as
efficient algorithms that determine maximum sets of disjoint paths of order $k$
and minimum sets of vertices intersecting all paths of order $k$.
