On the Complexity of the K-way Vertex Cut Problem

TL;DR

This paper investigates the computational complexity of the K-way vertex cut problem, revealing its NP-completeness on bipartite graphs and establishing its equivalence to the Critical Node Problem on split graphs.

Contribution

It proves the NP-completeness of the K-way vertex cut problem on bipartite graphs and shows its equivalence to the Critical Node Problem on split graphs, extending understanding of its complexity.

Findings

NP-complete on bipartite graphs

Equivalent to the Critical Node Problem on split graphs

Extends complexity understanding of the problem

Abstract

The K-way vertex cut problem} consists in, given a graph G, finding a subset of vertices of a given size, whose removal partitions G into the maximum number of connected components. This problem has many applications in several areas. It has been proven to be NP-complete on general graphs, as well as on split and planar graphs. In this paper, we enrich its complexity study with two new results. First, we prove that it remains NP-complete even when restricted on the class of bipartite graphs. This is unlike what it is expected, given that the K-way vertex cut problem is a generalization of the Maximum Independent set problem which is polynomially solvable on bipartite graphs. We also provide its equivalence to the wellknown problem, namely the Critical Node Problem (CNP), On split graphs. Therefore, any solving algorithm for the CNP on split graphs is a solving algorithm for the K-way vertex cut problem and vice versa.