Characterization of Well-Totally Dominated Trees

Let $G$ be a graph with no isolated vertices. A set of vertices $S$ is a total dominating set (TDS) if every vertex in $G$ is adjacent to at least one vertex in $S$. We say $G$ is well-totally dominated (WTD) if every minimal TDS has the same size. In this paper, we present two characterizations of well-totally dominated trees, one being descriptive and the other being constructive. In particular, our characterizations imply that it takes only polynomial time to verify whether a given tree is WTD.