Complexity and Algorithm for the Matching vertex-cutset Problem

TL;DR

This paper studies the complexity of finding specific structured vertex cutsets in graphs, proving NP-completeness for matching vertex-cutsets, providing approximation algorithms, and establishing bounds for plane graphs.

Contribution

It proves NP-completeness for the matching vertex-cutset problem, offers a 2-approximation algorithm, and determines tight bounds for plane graphs.

Findings

Matching vertex-cutset decision problem is NP-complete.

A 2-approximation algorithm exists for minimum matching vertex-cutset.

Every plane graph (except K2 and K4) has a matching vertex-cutset of size at most three.

Abstract

In 1985, Chv\'{a}tal introduced the concept of star cutsets as a means to investigate the properties of perfect graphs, which inspired many researchers to study cutsets with some specific structures, for example, star cutsets, clique cutsets, stable cutsets. In recent years, approximation algorithms have developed rapidly, the computational complexity associated with determining the minimum vertex cut possessing a particular structural property have attracted considerable academic attention. In this paper, we demonstrate that determining whether there is a matching vertex-cutset in $H$ with size at most $k$, is $\mathbf{NP}$-complete, where $k$ is a given positive integer and $H$ is a connected graph. Furthermore, we demonstrate that for a connected graph $H$, there exists a $2$-approximation algorithm in $O(nm^2)$ for us to find a minimum matching vertex-cutset. Finally, we show that every plane graph $H$ satisfying $H\not\in\{K_2, K_4\}$ contains a matching vertex-cutset with size at most three, and this bound is tight.