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a b s t r a c tIn this paper, we continue the study of locating-total domination in graphs, introduced by Haynes et al. [T.W. Haynes, M.A. Henning, J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics 154 (8) (2006) 1293-1300]. A total dominating set S in a graph G = (V , E) is a locating-total dominating set of G if, for every pair of distinct vertices u andThe minimum cardinality of a locating-total dominating set is the locating-total domination number γ L t (G). We show that, for a tree T of order n ≥ 3 with l leaves and s support vertices, n+l+1Moreover, we constructively characterize the extremal trees achieving these bounds.

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The inflation G I of a graph G with n(G) vertices and m(G) edges is obtained from G by replacing every vertex of degree d of G by a clique K d. A set S of vertices in a graph G is a paired dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired domination number γ p (G) is the minimum cardinality of a paired dominating set of G. In this paper, we show that if a graph G has a minimum degree δ(G) ≥ 2, then n(G) ≤ γ p (G I) ≤ 4m(G) δ(G)+1 , and the equality γ p (G I) = n(G) holds if and only if G has a perfect matching. In addition, we present a linear time algorithm to compute a minimum paired-dominating set for an inflation tree.