Minimum k-path vertex cover

TL;DR

This paper studies the minimum size of vertex sets covering all paths of length k in a graph, proving NP-hardness for general graphs, but providing bounds and exact values, especially for trees and specific cases.

Contribution

It establishes NP-hardness of the k-path vertex cover problem for all k ≥ 2 and offers new bounds and exact solutions, including a specific upper bound for  in general graphs.

Findings

NP-hardness for general graphs when k  2

Linear-time solution for trees

Upper bound   (2n + m)/6 for all graphs

Abstract

A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by \psi_k(G) the minimum cardinality of a k-path vertex cover in G. It is shown that the problem of determining \psi_k(G) is NP-hard for each k \geq 2, while for trees the problem can be solved in linear time. We investigate upper bounds on the value of \psi_k(G) and provide several estimations and exact values of \psi_k(G). We also prove that \psi_3(G) \leq (2n + m)/6, for every graph G with n vertices and m edges.