A hypergraph is conformal if it is the family of maximal cliques of a graph. In this paper we are interested in the problem of determining when is the family of minimal transversal of maximal cliques of a graph conformal. Such graphs are called clique dually conformal (CDC for short). 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. 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.

翻译：超图若为某图的极大团族，则称其共形。本文关注的问题是何条件下图的极大团的最小横贯族为共形的，此类图称为团对偶共形图（简称CDC图）。作为主要结果，我们完全刻画了无三角形图族和分裂图族中的CDC图。两种刻画均能导出多项式时间识别算法。此外，我们证明CDC图类在代换运算下封闭，其强意义表现为：将图$H$代换到图$G$的某顶点上所得图为CDC图当且仅当$G$与$H$均为CDC图。