A characterization of trees with equal total domination and paired-domination numbers

Let G = (V,E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to at least one vertex in S. A total dominating set S ⊆ V is a paired-dominating set if the induced subgraph G[S] has at least one perfect matching. The paired-domination number γpr(G) is the minimum cardinality of a paired-domination set of G. In this paper, we provide a constructive characterization of those trees with equal total domination and paired-domination numbers, and of those trees for which the paired domination number is twice the matching number. ∗ Research of the first and second authors supported by the National Nature Science Foundation of China under grant 10101010 and the Young Science Foundation of Shanghai Education Committee under grant 01QN6262. † Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal. 32 ERFANG SHAN, LIYING KANG AND MICHAEL HENNING