Matching Cut in Graphs with Large Minimum Degree

In a graph, a matching cut is an edge cut that is a matching. M atching C ut is the problem of deciding whether or not a given graph has a matching cut, which is known to be 𝖭𝖯 -complete. While M atching C ut 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 , M atching C ut is 𝖭𝖯 -complete in the class of graphs with minimum degree c and this restriction of M atching C ut 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 , M atching C ut remains 𝖭𝖯 -complete in the class of n -vertex (bipartite) graphs with unbounded minimum degree δ >n^1-ϵ . We give an exact branching algorithm to solve M atching C ut for graphs with minimum degree δ≥ 3 in O^*(λ ^n) time, where λ is the positive root of the polynomial x^δ +1-x^δ-1 . Despite the hardness results, this is a very fast exact exponential-time algorithm for M atching C ut 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 , M atching C ut is solvable in polynomial time for graphs with large minimum degree δ≥1/cn-c^' .