Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs
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A hypergraph H is said to be 1-Sperner if for every two hyperedges the smallest of their two set differences is of size one. The authors introduced this concept in 2015 and completely characterized this class of hypergraphs via a decomposition theorem. Here, 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 the decomposition theorem for 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.
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