Abstract
In a graph, a matching cut is an edge cut that is a matching. Matching Cut, which is known to be NP-complete, is the problem of deciding whether or not a given graph G has a matching cut. In this paper we show that Matching Cut admits a quadratic-vertex kernel for the parameter distance to cluster and a linear-vertex kernel for the parameter distance to clique. We further provide an O∗(2dc(G))-time and an O∗(2dc¯(G))-time FPT algorithm for Matching Cut, where dc(G) and dc¯(G) are the distance to cluster and distance to co-cluster, respectively. We also improve the running time of the best known branching algorithm to solve Matching Cut from O∗(1.4143n) to O∗(1.3071n) where n is the number of vertices in G. Moreover, we point out that, unless NP⊆coNP∕poly, Matching Cut does not admit a polynomial kernel when parameterized simultaneously by treewidth, the number of edges crossing the cut, and the maximum degree of G.
Highlights
In a graph G = (V, E), a cut is a partition V = A ∪ ̇ B of the vertex set into disjoint, nonempty sets A and B, written (A, B)
Exact exponential algorithms for Matching Cut on graphs without any restriction have been recently considered by Kratsch and Le [14] who provided the first exact branching algorithm for Matching Cut running in time O∗(1.4143n)1, and a single-exponential algorithm of running time 2τ(G)O(n2), where τ (G) is the vertex cover number
We provided three algorithms for Matching Cut: an exact exponential algorithm of running time O∗(1.3803n), a fixed-parameter algorithm of running time 2dc(G)O(n2) where dc(G) is the distance to cluster number, and a fixed-parameter algorithm of running time 2dc(G)O(nm) where dc(G) is the distance to co-cluster number
Summary
Exact exponential algorithms for Matching Cut on graphs without any restriction have been recently considered by Kratsch and Le [14] who provided the first exact branching algorithm for Matching Cut running in time O∗(1.4143n), and a single-exponential algorithm of running time 2τ(G)O(n2), where τ (G) is the vertex cover number. We show that Matching Cut can be solved by single-exponential algorithms running in time 2dc(G)O(n2) and 2dc(G)O(nm), respectively, where dc(G) is the distance to cluster and dc(G) is the distance to co-cluster This improves upon the FPT algorithms for Matching Cut with running time 2τ(G)O(n2) [14], where τ (G) ≥ max{dc(G), dc(G)} is the vertex cover number of G which can be much larger than dc(G) and dc(G). For the basic notions of parameterized complexity we refer to [8]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
