Trees with equal total domination and paired-domination numbers - Henning, Michael A. (Utilitas Mathematica, 2006)
A total dominating set of a graph G with no isolated vertex is a set S of vertices of G such that every vertex is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. A paired-dominating set of G is a dominating set of vertices whose induced subgraph has a perfect matching. The paired-domination number of G is the minimum cardinality of a paired-dominating set of G. Since every paired-dominating set is a total dominating set, the total domination number is bounded above by the paired-domination number. We give a constructive characterization of the trees with equal total domination and paired-domination numbers. Our characterization uses labelings of the vertices that indicates the roles each vertex plays in the sets associated with both parameters.