A Parameterized Algorithm for Mixed-Cut

The classical Menger’s theorem states that in any undirected (or directed) graph G, given a pair of vertices s and t, the maximum number of vertex (edge) disjoint paths is equal to the minimum number of vertices (edges) needed to disconnect s from t. This min-max result can be turned into a polynomial time algorithm to find the maximum number of vertex (edge) disjoint paths as well as the minimum number of vertices (edges) needed to disconnect s from t. In this paper we study a mixed version of this problem, called Mixed-Cut, where we are given an undirected graph G, vertices s and t, positive integers k and l and the objective is to test whether there exist a k sized vertex set S ⊆ V (G) and an l sized edge set F ⊆ E(G) such that deletion of S and F from G disconnects from s and t. Apart from studying a generalization of classical problem, one of our main motivations for studying this problem comes from the fact that this problem naturally arises as a subproblem in the study of several graph editing (modification) problems. We start with a small observation that this problem is NP-complete and then study this problem, in fact a much stronger generalization of this, in the realm of parameterized complexity. In particular we study the Mixed Multiway Cut-Uncut problem where along with a set of terminals T , we are also given an equivalence relation R on T , and the question is whether we can delete at most k vertices and at most l edges such that connectivity of the terminals in the resulting graph respects R. Our main result is a fixed parameter algorithm for Mixed Multiway Cut-Uncut using the method of recursive understanding introduced by Chitnis