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 c> 1 , Matching Cut is NP-complete in the class of graphs with minimum degree c and this restriction of Matching Cut has no subexponential-time algorithm in the number of vertices unless the Exponential-Time Hypothesis fails. We also show that, for any given constant ϵ> 0 , Matching Cut remains NP-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree δ> n1-ϵ. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree δ≥ 3 in O∗(λn) time, where λ is the positive root of the polynomial xδ+1- xδ- 1. Despite the hardness results, this is a very fast exact exponential-time algorithm for Matching Cut on graphs with large minimum degree; for instance, the running time is O∗(1. 0099 n) on graphs with minimum degree δ≥ 469. Complementing our hardness results, we show that, for any two fixed constants 1 < c< 4 and c′≥ 0 , Matching Cut is solvable in polynomial time for graphs with large minimum degree δ≥1cn-c′.
All Science Journal Classification (ASJC) codes
- Computer Science(all)
- Computer Science Applications
- Applied Mathematics