In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 199–206). A paired-dominating set of a graph G with no isolated vertex is a dominating set S of vertices whose induced subgraph has a perfect matching. The set S is called a differentiating-paired dominating set if for every pair of distinct vertices u and v in V(G), N[u]∩S≠N[v]∩S, where N[u] denotes the set consisting of u and all vertices adjacent to u. In this paper, we provide a constructive characterization of trees that do not have a differentiating-paired dominating set.
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