On structural parameterizations of the matching cut problem

Abstract

In an undirected graph, a matching cut is a partition of vertices into two sets such that the edges across the sets induce a matching. The matching cut problem is the problem of deciding whether a given graph has a matching cut. The matching cut problem can be expressed using a monadic second-order logic (MSOL) formula and hence is solvable in linear time for graphs with bounded tree-width. However, this approach leads to a running time of f(ϕ, t) nO(1), where ϕ is the length of the MSOL formula, t is the tree-width of the graph and n is the number of vertices of the graph. In [Theoretical Computer Science, 2016], Kratsch and Le asked to give a single exponential algorithm for the matching cut problem with tree-width alone as the parameter. We answer this question by giving a 2 O(t)nO(1) time algorithm. We also show the tractability of the matching cut problem when parameterized by neighborhood diversity and other structural parameters.