2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/132799A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that each component of which is a star. Recently, Hartnell and Rall studied a family $\mathscr{U}$ of graphs satisfying the property that every star-factor of a member graph has the same number of edges. They determined the family $\mathscr{U}$ when the girth is at least five. In this paper, we investigate the family of graphs with girth three and determine all members of this family.Combinatorics05C69, 05C70text