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Total domination subdivision numbers of graphs. (English) Zbl 1065.05070

Summary: 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.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C35 Extremal problems in graph theory
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