The k-path vertex cover: general bounds and chordal graphs

TL;DR

This paper investigates bounds on the size of k-path vertex covers in general, chordal, and planar graphs, providing estimates based on graph parameters and exploring specific graph classes.

Contribution

It introduces new bounds for the k-path vertex cover number, especially for chordal and planar graphs, expanding understanding of this problem in various graph classes.

Findings

Provides upper bounds on _k(G) based on degrees and graph size

Analyzes the problem specifically for chordal graphs

Extends the study to planar graphs

Abstract

For an integer $k\ge 3$, a $k$-path vertex cover of a graph $G=(V,E)$ is a set $T\subseteq V$ that shares a vertex with every path subgraph of order $k$ in $G$. The minimum cardinality of a $k$-path vertex cover is denoted by $\psi_k(G)$. We give estimates -- mostly upper bounds -- on $\psi_k(G)$ in terms of various parameters, including vertex degrees and the number of vertices and edges. The problem is also considered on chordal graphs and planar graphs.