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Domination subdivision numbers. (English) Zbl 1006.05042

A set \(S\) of vertices of a graph \(G\) is a dominating set if every vertex of \(V(G)-S\) is adjacent to some vertex in \(S\). The domination number \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\), and the domination subdivision number \(\text{sd}_{\gamma}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the domination number. In 2000, Arumugam defined the domination subdivision number, and he conjectured that \(1\leq \text{sd}_{\gamma}(G)\leq 3\) for any graph \(G\). In this paper, the authors construct a counterexample to this conjecture, and they show that \(\text{sd}_{\gamma}(G)\leq\gamma(G)+1\) for any graph \(G\) without isolated vertices. Furthermore, they give constant upper bounds on \(\text{sd}_{\gamma}(G)\) for several families of graphs. Finally, the authors present some interesting open problems and conjectures.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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