1. Conformality of minimal transversals of maximal cliquesEndre Boros, Vladimir Gurvich, Martin Milanič, Dmitry Tikhanovsky, Yushi Uno, 2026, original scientific article Abstract: Given a hypergraph ℋ, the dual hypergraph of ℋis the hypergraph of all minimal transversals of ℋ. A hypergraph is conformal if it is the family of maximal cliques of a graph. In a recent work, Boros, Gurvich, Milanič, and Uno (Journal of Graph Theory, 2025) studied conformality of dual hypergraphs and proved several results related to this property, leading in particular to a polynomial-time algorithm for recognizing graphs in which, for any fixed k, all minimal transversals of maximal cliques have size at most k. In this follow-up work, we provide a novel aspect to the study of graph clique transversals, by considering the dual conformality property from the perspective of graphs. More precisely, we study graphs for which the family of minimal transversals of maximal cliques is conformal. Such graphs are called clique dually conformal (CDC for short). It turns out that the class of CDC graphs is a rich generalization of the class of P4-free graphs. As our main results, we completely completely characterize CDC graphs within the families of triangle-free graphs and split graphs. Both characterizations lead to polynomial-time recognition algorithms. Generalizing the fact that every P4-free graph is CDC, we also show that the class of CDC graphs is closed under substitution, in the strong sense that substituting a graph H for a vertex of a graph G results in a CDC graph if and only if both G and H are CDC.
Keywords: maximal clique, minimal transversal, conformal hypergraph, triangle-free graph, split graph
Published in RUP: 16.12.2025; Views: 263; Downloads: 2
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2. Conformal hypergraphs : duality and implications for the upper clique transversal problemEndre Boros, Vladimir Gurvich, Martin Milanič, Yushi Uno, 2025, original scientific article Abstract: Given a hypergraph H, the dual hypergraph of H is the hypergraph of all minimal transversals of H. The dual hypergraph is always Sperner, that is, no hyperedge contains another. A special case of Sperner hypergraphs are the conformal Sperner hypergraphs, which correspond to the families of maximal cliques of graphs. All these notions play an important role in many fields of mathematics and computer science, including combinatorics, algebra, database theory, etc. In this paper we study conformality of dual hypergraphs and prove several results related to the problem of recognizing this property. In particular, we show that the problem is in co-NP and can be solved in polynomial time for hypergraphs of bounded dimension. In the special case of dimension 3, we reduce the problem to 2-Satisfiability. Our approach has an implication in algorithmic graph theory: we obtain a polynomial-time algorithm for recognizing graphs in which all minimal transversals of maximal cliques have size at most k, for any fixed k.
Keywords: conformal hypergraph, dual hypergraph, hypergraph, maximal clique, upper clique transversal
Published in RUP: 25.09.2025; Views: 393; Downloads: 4
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4. Sprague-Grundy function of matroids and related hypergraphsEndre Boros, Vladimir Gurvich, Nhan Bao Ho, Kazuhisa Makino, Peter Muršič, 2019, original scientific article Keywords: matroid, self-dual matroid, impartial game, Sprague-Grundy function, Nim, hypergraph Nim, JM hypergraph
Published in RUP: 11.02.2020; Views: 2738; Downloads: 105
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6. Sprague-Grundy function of symmetric hypergraphsEndre Boros, Vladimir Gurvich, Nhan Bao Ho, Kazuhisa Makino, Peter Muršič, 2019, original scientific article Keywords: impartial games, Sprague-Grundy function, Nim, hypergraph Nim, JM formula, JM hypergraph, symmetric hypergraph, transversal-free hypergraph
Published in RUP: 22.05.2019; Views: 3395; Downloads: 134
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7. Linear separation of connected dominating sets in graphsNina Chiarelli, Martin Milanič, 2019, original scientific article Keywords: connected dominating set, connected domination, connected-domishold graph, forbidden induced subgraph characterization, split graph, chordal graph, minimal cutset, minimal separator, 1-Sperner hypergraph, threshold hypergraph, threshold Boolean function, polynomial-time algorithm
Published in RUP: 04.04.2019; Views: 4921; Downloads: 167
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8. Decomposing 1-Sperner hypergraphs, with applications to graphs, Journée-séminaire de combinatoire (équipe CALIN du LIPN, Université Paris-Nord, Villetaneuse), 11. 9. 2018Martin Milanič, 2018, invited lecture at foreign university Keywords: 1-Sperner hypergraph, threshold hypergraph, decomposition, threshold graph, clique-width
Published in RUP: 30.09.2018; Views: 4714; Downloads: 26
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9. Decomposing 1-Sperner hypergraphs, with applications to graphs, Séminaires du Pôle 2 : Optimisation combinatoire, algorithmique", LAMSADE, Université Paris-Dauphine, 17. 9. 2018Martin Milanič, 2018, invited lecture at foreign university Keywords: 1-Sperner hypergraph, threshold hypergraph, decomposition, threshold graph, clique-width
Published in RUP: 30.09.2018; Views: 3751; Downloads: 29
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