Domination, matching and transversal numbers for Berge-$G$ hypergraphs

TL;DR

This paper investigates domination, matching, and transversal numbers in Berge-G hypergraphs, a generalization of power hypergraphs, revealing new relationships among these parameters.

Contribution

It introduces and analyzes dilation hypergraphs of a graph G, extending previous work on generalized power hypergraphs with new bounds and relations.

Findings

Established bounds for domination, matching, and transversal numbers.

Generalized results to dilation hypergraphs of G.

Connected these parameters in the context of Berge-G hypergraphs.

Abstract

Let $G=(V(G),E(G))$ be a graph and $H=(V(H),E(H))$ be a hypergraph. The hypergraph $H$ is a {\it Berge-G} if there is a bijection $f : E(G) \mapsto E(H)$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We define {\it dilations of $G$} as a particular subfamily of not necessarily uniform Berge-$G$ hypergraphs. We examine domination, matching and transversal numbers and some relation between these parameters in that family of hypergraphs. Our work generalizes previous results concerning generalized power hypergraphs.