Geodesic
Trees with unique minimum total dominating sets
Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 2, pp. 233-246

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A set S of vertices of a graph G is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.
Keywords: domination, total domination 
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Haynes, Teresa; Henning, Michael. Trees with unique minimum total dominating sets. Discussiones Mathematicae. Graph Theory, Tome 22 (2002) no. 2, pp. 233-246. http://geodesic.mathdoc.fr/item/DMGT_2002_22_2_a2/