Geodesic
Total outer-connected domination in trees
Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 3, pp. 377-383

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Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by γ_tc(G), is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then γ_tc(T) ≥ ⎡2n/3⎤. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.
Keywords: total outer-connected domination number, domination number 
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Cyman, Joanna. Total outer-connected domination in trees. Discussiones Mathematicae. Graph Theory, Tome 30 (2010) no. 3, pp. 377-383. http://geodesic.mathdoc.fr/item/DMGT_2010_30_3_a1/