Work

Conformal Hypergraphs: Duality and Implications for the Upper Clique Transversal Problem

Year: 2025

Type: article

Abstract: ABSTRACT Given a hypergraph , the dual hypergraph of is the hypergraph of all minimal transversals of . 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, and so on. Motivated by a question related to clique transversals in graphs, we study in this paper 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 that it can be solved in polynomial time for hypergraphs of bounded dimension. For dimension 3, we show that the problem can be reduced to 2‐Satisfiability. Our approach has an application 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 , for any fixed .

Source: Journal of Graph Theory

DOI: 10.1002/jgt.23238

Authors: Endre Boros, Vladimir Gurvich, Martin Milanič, Yushi Uno

Institutions: Rutgers, The State University of New Jersey, National Research University Higher School of Economics, University of Primorska, Osaka Metropolitan University

Language: English

Topic: Advanced Graph Theory Research

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,150

Funders: National Research University Higher School of Economics, Japan Society for the Promotion of Science