Complexity and Algorithm for the Matching vertex-cutset Problem
Hengzhe Li, Qiong Wang, Jianbing Liu, Yanhong Gao
TL;DR
The paper investigates the matching vertex-cutset problem, asking for a matching $M$ in a connected graph $H$ such that $H-V(M)$ is disconnected or trivial. It proves $ extbf{NP}$-completeness for graphs not equal to $K_{2n}$ or $K_{n,n}$ via a reduction from the edge dominating set problem, and presents a $2$-approximation algorithm running in $O(nm^{2})$ to find a minimum matching vertex-cutset. It further analyzes plane graphs, showing that any connected plane graph $H\notin\{K_2,K_4\}$ satisfies $ kappa_M(H)\le 3$ with a tight bound, and discusses implications for maximal planar graphs. Overall, the work contributes to complexity and algorithmic approaches for structure-constrained connectivity in graphs, with potential applications in network reliability and graph drawing.
Abstract
In 1985, Chvátal 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.