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 $k$.

翻译：给定一个超图 $\mathcal{H}$，其对偶超图是 $\mathcal{H}$ 所有极小横贯的集合。对偶超图总是Sperner的，即没有一条超边包含另一条。Sperner超图的一个特例是一致Sperner超图，它对应于图的极大团族。这些概念在数学和计算机科学的许多领域中扮演着重要角色，包括组合学、代数、数据库理论等。本文研究了对偶超图的一致性，并证明了与识别该性质问题相关的若干结果。特别地，我们证明了该问题属于co-NP，并且对于有界维数的超图可以在多项式时间内求解。在维数为$3$的特殊情况下，我们将该问题归约为$2$-可满足性问题。我们的方法在图论算法中具有应用：对于任意固定的$k$，我们得到了一个多项式时间算法，用于识别所有极大团极小横贯大小不超过$k$的图。