Geodesic
Total domination subdivision numbers of graphs
Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 3, pp. 457-467

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A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. First we establish bounds on the total domination subdivision number for some families of graphs. Then we show that the total domination subdivision number of a graph can be arbitrarily large.
Keywords: total domination number, total domination subdivision number 
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Haynes, Teresa; Henning, Michael; Hopkins, Lora. Total domination subdivision numbers of graphs. Discussiones Mathematicae. Graph Theory, Tome 24 (2004) no. 3, pp. 457-467. http://geodesic.mathdoc.fr/item/DMGT_2004_24_3_a8/