On Blockers and Transversals of Maximum Independent Sets in Co-Comparability Graphs

In this paper, we consider the following two problems: (i) Deletion Blocker($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that $\alpha(G-S) \leq \alpha(G)-d$, that is the independence number of $G$ decreases by at least $d$ after having removed the vertices from $S$; (ii) Transversal($\alpha$) where we are given an undirected graph $G=(V,E)$ and two integers $k,d\geq 1$ and ask whether there exists a subset of vertices $S\subseteq V$ with $|S|\leq k$ such that for every maximum independent set $I$ we have $|I\cap S| \geq d$. We show that both problems are polynomial-time solvable in the class of co-comparability graphs by reducing them to the well-known Vertex Cut problem. Our results generalize a result of [Chang et al., Maximum clique transversals, Lecture Notes in Computer Science 2204, pp. 32-43, WG 2001] and a recent result of [Hoang et al., Assistance and interdiction problems on interval graphs, Discrete Applied Mathematics 340, pp. 153-170, 2023]

Dichotomies for Maximum Matching Cut: H-Freeness, Bounded Diameter, Bounded Radius

The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. 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 2 and to (P?+sP?)-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 2P?-free quadrangulated graphs of diameter 3 and radius 2 and for subcubic line graphs of triangle-free graphs. In this way, we obtain full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs

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 P_r-free graphs. This addresses a question of Bouquet and Picouleau (arXiv, 2020). For all three problems, we give state-of-the-art summaries of their computational complexity for H-free graphs

Dichotomies for Maximum Matching Cut: HH-Freeness, Bounded Diameter, Bounded Radius

The (Perfect) Matching Cut problem is to decide if a graph $G$ has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of $G$. 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 $2$ and to $(P_6 + sP_2)$-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 $2P_3$-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 $H$-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

Matching cuts in graphs of high girth and H-free graphs

International audienceThe (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed d ≥ 1, is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for H-subgraph-free graphs where H is any finite set of graphs