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

TL;DR

This paper investigates the properties of conformal dual hypergraphs, providing complexity results and algorithms for recognizing conformality, with implications for the upper clique transversal problem in graph theory.

Contribution

It establishes the complexity of recognizing conformal dual hypergraphs and offers polynomial-time algorithms for special cases, including bounded dimension and fixed transversal size.

Findings

Recognition problem is in co-NP.

Polynomial-time algorithm for hypergraphs of bounded dimension.

Reduction to 2-Satisfiability for dimension 3.

Abstract

Given a hypergraph $\mathcal{H}$, the dual hypergraph of $\mathcal{H}$ is the hypergraph of all minimal transversals of $\mathcal{H}$. 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 $k$.