312 research outputs found

    Proper circular arc graphs as intersection graphs of paths on a grid

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    In this paper we present a characterisation, by an infinite family of minimal forbidden induced subgraphs, of proper circular arc graphs which are intersection graphs of paths on a grid, where each path has at most one bend (turn)

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

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    In this paper, we consider the following two problems: (i) Deletion Blocker(α\alpha) where we are given an undirected graph G=(V,E)G=(V,E) and two integers k,d≥1k,d\geq 1 and ask whether there exists a subset of vertices S⊆VS\subseteq V with ∣S∣≤k|S|\leq k such that α(G−S)≤α(G)−d\alpha(G-S) \leq \alpha(G)-d, that is the independence number of GG decreases by at least dd after having removed the vertices from SS; (ii) Transversal(α\alpha) where we are given an undirected graph G=(V,E)G=(V,E) and two integers k,d≥1k,d\geq 1 and ask whether there exists a subset of vertices S⊆VS\subseteq V with ∣S∣≤k|S|\leq k such that for every maximum independent set II we have ∣I∩S∣≥d|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]

    A note on r-equitable k-colorings of trees

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    A graph G = (V, E) is r-equitably k-colorable if there exists a partition of V into k independent sets V¹, V², ... , Vk such that | |Vi| − |Vj| | ≤ r for all i, j ∈ {1, 2, ... , k}. In this note, we show that if two trees T¹ and T² of order at least two are r-equitably k-colorable for r ≥ 1 and k ≥ 3, then all trees obtained by adding an arbitrary edge between T¹ and T² are also r-equitably k-colorable

    Finding Matching Cuts in H-Free Graphs

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    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: H-Freeness, Bounded Diameter, Bounded Radius

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    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
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