1. Treewidth is NP-complete on cubic graphsHans L. Bodlaender, Édouard Bonnet, Lars Jaffke, Dušan Knop, Paloma T. Lima, Martin Milanič, Sebastian Ordyniak, Sukanya Pandey, Ondřej Suchý, 2025, original scientific article Abstract: The concept of avoidable paths in graphs was introduced by Beisegel, Chudnovsky, Gurvich, Milanič, and Servatius in 2019 as a common generalization of avoidable vertices and simplicial paths. In 2020, Bonamy, Defrain, Hatzel, and Thiebaut proved that every graph containing an induced path of order k also contains an avoidable induced path of the same order. They also asked whether one could generalize this result to other avoidable structures, leaving the notion of avoidability up to interpretation. In this paper, we address this question: we specify the concept of avoidability for arbitrary graphs equipped with two terminal vertices. We provide both positive and negative results, some of which are related to a recent work by Chudnovsky, Norin, Seymour, and Turcotte in 2024. We also discuss several open questions.
Keywords: treewidth, cubic graph, NP'completeness
Published in RUP: 18.12.2025; Views: 146; Downloads: 0
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2. On {k}-Roman graphsKenny Bešter Štorgel, Nina Chiarelli, Lara Fernández, J. Pascal Gollin, Claire Hilaire, Valeria Alejandra Leoni, Martin Milanič, 2025, published scientific conference contribution Abstract: For a positive integer k, a {k}-Roman dominating function of a graph G = (V, E) is a function f : V → {0, 1, . . . , k} satisfying f (N(v)) ≥ k for each vertex v ∈ V with f (v) = 0. Every graph G satisfies γ{Rk}(G) ≤ kγ(G), where γ{Rk}(G) denotes the minimum weight of a {k}-Roman dominating function of G and γ(G) is the domination number of G. In this work we study graphs for which the equality is reached, called {k}-Roman graphs. This extends the concept of {k}-Roman trees studied by Wang et al. in 2021 to gen- eral graphs. We prove that for every k ≥ 3, the problem of recognizing {k}-Roman graphs is NP-hard, even when restricted to split graphs. We provide partial answers to the question of which split graphs are {2}-Roman: we characterize {2}-Roman split graphs that can be decomposed with respect to the split join operation into two smaller split graphs and classify the {k}-Roman property within two specific families of split graphs that are prime with respect to the split join operation: suns and their complements.
Keywords: graph domination, {k}-Roman domination, {k}-Roman graph, split graph, split join, NP-completeness
Published in RUP: 16.12.2025; Views: 205; Downloads: 2
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7. A note on domination and independence-domination numbers of graphsMartin Milanič, 2013, published scientific conference contribution Abstract: Vizing's conjecture is true for graphs ▫$G$▫ satisfying ▫$\gamma^i(G) = \gamma(G)$▫, where ▫$\gamma(G)$▫ is the domination number of a graph ▫$G$▫ and ▫$\gamma^i(G)$▫ is the independence-domination number of ▫$G$▫, that is, the maximum, over all independent sets ▫$I$▫ in ▫$G$▫, of the minimum number of vertices needed to dominate ▫$I$▫. The equality ▫$\gamma^i(G) = \gamma(G)$▫ is known to hold for all chordal graphs and for chordless cycles of length ▫$0 \pmod{3}$▫. We prove some results related to graphs for which the above equality holds. More specifically, we show that the problems of determining whether ▫$\gamma^i(G) = \gamma(G) = 2$▫ and of verifying whether ▫$\gamma^i(G) \ge 2$▫ are NP-complete, even if ▫$G$▫ is weakly chordal. We also initiate the study of the equality ▫$\gamma^i = \gamma$▫ in the context of hereditary graph classes and exhibit two infinite families of graphs for which ▫$\gamma^i < \gamma$▫.
Keywords: Vizing's conjecture, domination number, independence-domination number, weakly chordal graph, NP-completeness, hereditary graph class, IDD-perfect graph
Published in RUP: 15.10.2013; Views: 4754; Downloads: 133
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