A Survey on the k-Path Vertex Cover Problem

TL;DR

This survey reviews the current research on the $k$-path vertex cover problem, focusing on its theoretical foundations, algorithms, and known results for various graph classes.

Contribution

It provides a comprehensive overview of existing methods, bounds, and open problems related to the $k$-path vertex cover problem and its computational complexity.

Findings

Summarizes key algorithms and bounds for the $k$-path vertex cover problem.

Highlights open problems and future research directions.

Reviews complexity results for different graph classes.

Abstract

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.