A hypergraph is said to be $1$-Sperner if for every two hyperedges the smallest of their two set differences is of size one. We present several applications of $1$-Sperner hypergraphs and their structure to graphs. In particular, we consider the classical characterizations of threshold and domishold graphs and use them to obtain further characterizations of these classes in terms of $1$-Spernerness, thresholdness, and $2$-asummability of their vertex cover, clique, dominating set, and closed neighborhood hypergraphs. Furthermore, we apply a decomposition property of $1$-Sperner hypergraphs to derive decomposition theorems for two classes of split graphs, a class of bipartite graphs, and a class of cobipartite graphs. These decomposition theorems are based on certain matrix partitions of the corresponding graphs, giving rise to new classes of graphs of bounded clique-width and new polynomially solvable cases of several domination problems.

中文翻译：

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通过 1-Sperner 超图表征和分解阈值、分裂和二部图的类别

如果对于每两个超边，它们的两个集合差中最小的大小为 1，则称超图为 $1$-Sperner。我们介绍了 $1$-Sperner 超图的几种应用及其在图中的结构。特别地，我们考虑了阈值图和域域图的经典特征，并使用它们来获得这些类的进一步特征，如顶点覆盖、集团、支配集和闭集的 $1$-Spernerness、阈值和 $2$-asummability邻域超图。此外，我们应用 $1$-Sperner 超图的分解性质来推导出两类分裂图、一类二部图和一类协二部图的分解定理。这些分解定理基于相应图的某些矩阵划分，