On the K\H{o}nig-Egerv\'ary Theorem for $k$-Paths

TL;DR

This paper extends the Konig-Egerve1ry theorem to $k$-paths, providing structural characterizations and efficient algorithms for recognizing and solving related path problems in specific graph classes.

Contribution

It offers complete structural descriptions of graphs where maximum disjoint $k$-paths equal minimum vertex covers for $k=3,4$, and for odd $k$ without small cycles, along with recognition algorithms.

Findings

Structural descriptions for $k=3,4$ graphs in al_k

Recognition algorithms for these graph classes

Algorithms for maximum disjoint $k$-paths and minimum vertex covers

Abstract

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 $k$.