Total restrained domination numbers of trees

For a given connected graphG= (V ,E), a setDtr ⊆ V (G) is a total restrained dominating set if it is dominating and both 〈Dtr〉 and 〈V (G)−Dtr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by tr(G). In this paper we characterize the trees with equal total and total restrained dominating numbers and give a lower bound on the total restrained dominating number of a tree T in terms of its order and the number of leaves of T . © 2007 Elsevier B.V. All rights reserved.