Geodesic
Trees with equal total domination and total restrained domination numbers
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 59-66

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For a graph G = (V,E), a set S ⊆ V(G) is a total dominating set if it is dominating and both 〈S〉 has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V(G) is a total restrained dominating set if it is total dominating and 〈V(G)-S〉 has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same.
Keywords: total domination number, total restrained domination number, tree 
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Chen, Xue-Gang; Shiu, Wai; Chen, Hong-Yu. Trees with equal total domination and total restrained domination numbers. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 59-66. http://geodesic.mathdoc.fr/item/DMGT_2008_28_1_a3/