Geodesic
Weakly connected domination subdivision numbers
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 109-119.

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A set D of vertices in a graph G = (V,E) is a weakly connected dominating set of G if D is dominating in G and the subgraph weakly induced by D is connected. The weakly connected domination number of G is the minimum cardinality of a weakly connected dominating set of G. The weakly connected domination subdivision number of a connected graph G is the minimum number of edges that must be subdivided (where each egde can be subdivided at most once) in order to increase the weakly connected domination number. We study the weakly connected domination subdivision numbers of some families of graphs.
Keywords: weakly connected domination number, weakly connected domination subdivision number 
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Raczek, Joanna. Weakly connected domination subdivision numbers. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 1, pp. 109-119. http://geodesic.mathdoc.fr/item/DMGT_2008_28_1_a7/

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