Characterizing and decomposing classes of threshold, split, and bipartite graphs via 1-Sperner hypergraphs

TL;DR

This paper explores the structure of 1-Sperner hypergraphs and their applications to characterizing and decomposing various classes of graphs, leading to new insights and polynomial algorithms for domination problems.

Contribution

It introduces new characterizations of threshold and split graphs using 1-Sperner hypergraphs and develops decomposition theorems for specific graph classes based on matrix partitions.

Findings

Characterizations of threshold and domishold graphs via 1-Sperner hypergraphs.

Decomposition theorems for split, bipartite, and cobipartite graphs.

Polynomial algorithms for certain domination problems.

Abstract

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.