Conformality of Minimal Transversals of Maximal Cliques

TL;DR

This paper investigates the class of graphs where minimal transversals of maximal cliques form a conformal hypergraph, providing characterizations and recognition algorithms for specific graph families, and establishing closure properties.

Contribution

It introduces the class of clique dually conformal (CDC) graphs, characterizes CDC graphs within triangle-free and split graphs, and proves closure under substitution.

Findings

CDC graphs generalize P4-free graphs.

Polynomial-time algorithms for recognizing CDC graphs in specific classes.

CDC class is closed under graph substitution.

Abstract

Given a hypergraph $H$, the dual hypergraph of $H$ is 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, Milani\v{c}, 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 all minimal transversals of maximal cliques have size at most $k$, for any fixed $k$. 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 $P_4$-free graphs. As our main results, we 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 $P_4$-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.