AbstractA set S ofvertices in a graphG is a total dominating set of G ifevery vertex ofG is adjacentto some vertex in S. We characterize the set ofvertices ofa tree that are contained in all, or inno, minimum total dominating sets ofthe tree. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Total domination; Total domination number; Tree 1. IntroductionLet G=(V;E) be a simple undirected graph with | V | =n. The open neighborhoodof v ∈ V is N(v)= { u ∈ V | uv ∈ E } . A set S ⊆ V is a dominating set of G ifeveryvertex in V − S is adjacent to a vertex of S. The domination number (G)istheminimum cardinality ofa dominating set ofG. A set S ⊆ V is a total dominating set,abbreviated TDS,ofG ifevery vertex inV is adjacent to a vertex in S. Every graphwithout isolated vertices has a TDS, since S =V is such a set. The total dominationnumber of G, denoted by t (G), is the minimum cardinality ofa TDS. A TDS ofcardinality t (G) is called a t (G)-set. ∗ Corresponding author.E-mail address: henning@nu.ac.za (M.A. Henning).