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 $${\mathsf {NP}}$$
NP
-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is $${\mathsf {NP}}$$
NP
-complete on graphs with minimum degree two. In this paper, we show that, for any given constant $$c>1$$
c
>
1
, Matching Cut is $${\mathsf {NP}}$$
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 $$\epsilon >0$$
ϵ
>
0
, Matching Cut remains $${\mathsf {NP}}$$
NP
-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree $$\delta >n^{1-\epsilon }$$
δ
>
n
1
-
ϵ
. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree $$\delta \ge 3$$
δ
≥
3
in $$O^*(\lambda ^n)$$
O
∗
(
λ
n
)
time, where $$\lambda$$
λ
is the positive root of the polynomial $$x^{\delta +1}-x^{\delta }-1$$
x
δ
+
1
-
x
δ
-
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 $$O^*(1.0099^n)$$
O
∗
(
1
.
0099
n
)
on graphs with minimum degree $$\delta \ge 469$$
δ
≥
469
. Complementing our hardness results, we show that, for any two fixed constants $$1< c <4$$
1
<
c
<
4
and $$c^{\prime }\ge 0$$
c
′
≥
0
, Matching Cut is solvable in polynomial time for graphs with large minimum degree $$\delta \ge \frac{1}{c}n-c^{\prime }$$
δ
≥
1
c
n
-
c
′
.
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