  A {\it star-factor} of a graph $G$ is a spanning subgraph of $G$ such that
each of its component is a star. Clearly, every graph without isolated vertices
has a star factor. A graph $G$ is called {\it star-uniform} if all star-factors
of $G$ have the same number of components. To characterize star-uniform graphs
was an open problem posed by Hartnell and Rall, which is motivated by the
minimum cost spanning tree and the optimal assignment problems. We use the
concepts of factor-criticality and domination number to characterize all
star-uniform graphs with the minimum degree at least two. Our proof is heavily
relied on Gallai-Edmonds Matching Structure Theorem.
