Geodesic
Perfect connected-dominant graphs
Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 159-162

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If D is a dominating set and the induced subgraph G(D) is connected, then D is a connected dominating set. The minimum size of a connected dominating set in G is called connected domination number γ_c(G) of G. A graph G is called a perfect connected-dominant graph if γ(H) = γ_c(H) for each connected induced subgraph H of G.We prove that a graph is a perfect connected-dominant graph if and only if it contains no induced path P₅ and induced cycle C₅.
Keywords: Connected domination, perfect connected-dominant graph 
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Zverovich, Igor. Perfect connected-dominant graphs. Discussiones Mathematicae. Graph Theory, Tome 23 (2003) no. 1, pp. 159-162. http://geodesic.mathdoc.fr/item/DMGT_2003_23_1_a10/