Matching Cut in Graphs with Large Minimum Degree

Abstract

In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP -complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is NP -complete on graphs with minimum degree two. In this paper, we show that, for any given constant (forumala presented), Matching Cut is NP -complete in the class of n-vertex (bipartite) graphs with minimum degree (forumala presented). We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree (forumala presented). This is a very fast exact exponential time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is (forumala presented). Complementing our hardness results, we show that, for any fixed constant (forumala presented), Matching Cut is solvable in polynomial time for graphs with very large minimum degree (forumala presented).