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

Abstract

Given a hypergraph $\mathscr{H}$, the dual hypergraph of $\mathscr{H}$ is the hypergraph of all minimal transversals of $\mathscr{H}$. 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, 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 $k$, for any fixed $k$.