The k-way vertex cut problem on bipartite graphs: Complexity results and algorithms

Abstract

We consider the k-way vertex cut problem that consists in finding a subset of vertices of a given cardinality, in a graph, whose removal partitions the graph into the maximum connected components. This problem has been proven to be NP-complete on general graphs, split and planar graphs. In this paper, we consider it on bipartite graphs and we show that it remains NP-complete even restricted on this class of graphs. However, for the subclass of bipartite-permutation graphs, we develop a polynomial-time algorithm using the dynamic programming approach for solving the problem. The algorithm runs in $O\left(n{K}^2\right)$ time and $O\left(nK\right)$ space, where $n$ is the graph order, and $K$ is the number of deleted vertices. We also extend our attention by considering vertex deletion costs, and we adapt the proposed dynamic program to the case where non-negative costs are associated to vertex deletion. The obtained algorithm is of time and space complexity $O\left(n^3\right)$ and $O\left(n^2\right)$, respectively.