The smallest one-realization of a given set

TL;DR

This paper refines the understanding of minimal hypergraph realizations of a set by identifying the least number of certain edges, advancing solutions to an open problem in hypergraph theory.

Contribution

It determines the minimum number of ${\

Findings

Identified the minimum number of ${\

Partially solved an open problem by Tuza and Voloshin (2008).

Abstract

In [The smallest one-realization of a given set, Electronic J. Combin. 19 (2012), $\sharp$P19], we determined the minimum number of vertices of one-realizations of a given finite set $S$, and constructed the corresponding mixed hypergraphs. In this paper, by finding some of their spanning sub-hypergraphs, we determine the minimum number of ${\cal D}$-deges (resp. ${\cal C}$-edges) of one-realizations of $S$. As a result, we partially solve an open problem proposed by Tuza and Voloshin in 2008.