The Smallest One-Realization of a Given Set III

In Zhao et al. (Electron J Combin 19: ({sharp}) P19, 2012), 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 ({mathcal{D}}) -deges (resp. ({mathcal{C}}) -edges) of one-realizations of S. As a result, we partially solve an open problem proposed by Tuza and Voloshin (Bolyai Society Mathematical Studies, vol. 17, pp. 235–255. Springer, Berlin, 2008).