Matching Cut in Graphs with Large Minimum Degree

Abstract

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 ′ .

The Perfect Matching Cut Problem Revisited

“…A matching cut is an edge cut that is a (possibly empty) matching. Another way to define matching cuts is as follows; see [8,12]: a cut (X, Y ) is a matching cut if and only if each vertex in X has at most one neighbor in Y and each vertex in Y has at most one neighbor in X. matching cut (mc) is the problem of deciding if a given graph admits a matching cut and this problem has received much attention lately; see [7,10] for recent results. An interesting special case, where the edge cut E(X, Y ) is a perfect matching, was considered in [13].…”

“…Hardness results were obtained for further restricted graph classes such as bipartite graphs, planar graphs and graphs of bounded diameter (see [4,19,20]). Further graph classes in which mc is polynomial time solvable were identified, such as graphs of bounded tree-width, claw-free, hole-free and Ore-graphs (see [4,7,20]). FPT algorithms and kernelization for mc with respect to various parameters has been discussed in [1,2,10,11,17,18].…”

“…The current-best exact algorithm solving mc has a running time of O * (1.3280 n ) where n is the vertex number of the input graph [17]. Faster exact algorithms can be obtained for the case when the minimum degree is large [7]. The recent paper [10] addresses enumeration aspects of matching cuts.…”

A note on Matching-Cut in $P_t$-free Graphs

“…It was Chvátal [6] who initiated the study of matching-cut, the complexity problem of recognizing graphs admitting a matching-cut, showing that it is NP-complete, even for graphs with maximum degree at most four, yet in P for graphs with maximum degree at most three (unaware of Chvátal's result, Dunbar et al [7] formulated matching-cut, leaving it as an open problem that was repopularized in 2016 in [9]). The NP-hardness of matching-cut has since been shown to also hold for graphs with additional or other structural assumptions; see, for example, [2,5,10,11]. To keep this paper short, we refer the reader to [5,10] and references therein for a thorough discussion, including real-world applications.…”

“…The NP-hardness of matching-cut has since been shown to also hold for graphs with additional or other structural assumptions; see, for example, [2,5,10,11]. To keep this paper short, we refer the reader to [5,10] and references therein for a thorough discussion, including real-world applications.…”