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. While we do not settle the computational complexity status of recognizing this property, 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 $\mathcal{H}$ 的所有最小横贯的集合。对偶超图总是斯珀纳超图，即没有超边包含另一超边。斯珀纳超图的一个特例是一致斯珀纳超图，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发挥着重要作用，包括组合学、代数学、数据库理论等。本文研究对偶超图的一致性。虽然我们未解决识别该性质的计算复杂性状态，但证明了该问题属于 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启示意义：对于任意固定 $k$，我们获得了一个多项式时间算法，用于识别所有极大团的最小横贯大小至多为 $k$ 的图。