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}}-complete. While Matching Cut is trivial for graphs with minimum degree at most one, it is {mathsf {NP}}-complete on graphs with minimum degree two. In this paper, we show that, for any given constant c>1, Matching Cut is {mathsf {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, Matching Cut remains {mathsf {NP}}-complete in the class of n-vertex (bipartite) graphs with unbounded minimum degree delta >n^{1-epsilon }. We give an exact branching algorithm to solve Matching Cut for graphs with minimum degree delta ge 3 in O^*(lambda ^n) time, where lambda is the positive root of the polynomial x^{delta +1}-x^{delta }-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 delta ge 469. Complementing our hardness results, we show that, for any two fixed constants 1< c <4 and c^{prime }ge 0, Matching Cut is solvable in polynomial time for graphs with large minimum degree delta ge frac{1}{c}n-c^{prime }.
Highlights
In a graph G = (V, E), a cut is a partition V = X ∪̇ Y of the vertex set into disjoint, non-empty sets X and Y, written (X, Y)
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
For any constant c > 1, Matching Cut restricted on graphs with minimum degree c is -complete and, assuming the Exponential Time Hypothesis (ETH), cannot be solved in O∗(2o(n)) time for n-vertex graphs
Summary
A partition V = X ∪̇ Y of the vertex set of the graph G = (V, E) into disjoint, non-empty sets X and 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. In [6], Farley and Proskurowski studied matching cuts in the context of network applications. Matching cuts have been used by Araújo et al [1] in studying good edge-labellings in the context of WDM (Wavelength Division Multiplexing) networks. Matching Cut is trivial for graphs with minimum degree at most one. This paper considers the computational complexity of, as well as algorithms for the Matching Cut problem in graphs of large minimum degree
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