107 research outputs found
Dichotomies for Maximum Matching Cut: -Freeness, Bounded Diameter, Bounded Radius
The (Perfect) Matching Cut problem is to decide if a graph has a
(perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of
. Both Matching Cut and Perfect Matching Cut are known to be NP-complete,
leading to many complexity results for both problems on special graph classes.
A perfect matching cut is also a matching cut with maximum number of edges. To
increase our understanding of the relationship between the two problems, we
introduce the Maximum Matching Cut problem. This problem is to determine a
largest matching cut in a graph. We generalize and unify known polynomial-time
algorithms for Matching Cut and Perfect Matching Cut restricted to graphs of
diameter at most and to -free graphs. We also show that the
complexity of Maximum Matching Cut} differs from the complexities of Matching
Cut and Perfect Matching Cut by proving NP-hardness of Maximum Matching Cut for
-free graphs of diameter 3 and radius 2 and for line graphs. In this way,
we obtain full dichotomies of Maximum Matching Cut for graphs of bounded
diameter, bounded radius and -free graphs.Comment: arXiv admin note: text overlap with arXiv:2207.0709
Finding Matching Cuts in H-Free Graphs
The well-known NP-complete problem MATCHING CUT is to decide if a graph has a matching that is also an edge cut of the graph. We prove new complexity results for MATCHING CUT restricted to H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. We also prove new complexity results for two recently studied variants of MATCHING CUT, on H-free graphs. The first variant requires that the matching cut must be extendable to a perfect matching of the graph. The second variant requires the matching cut to be a perfect matching. In particular, we prove that there exists a small constant r>0 such that the first variant is NP-complete for Pr-free graphs. This addresses a question of Bouquet and Picouleau (The complexity of the Perfect Matching-Cut problem. CoRR, arXiv:2011.03318, (2020)). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs
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