Domination Subdivision Numbers in Graphs
A set S of vertices of a graph G = (V, E) is a dominating set if every vertex in V - S is adjacent to some vertex in 3. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(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 June 2000, Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. However, a counterexample to this conjecture given in  suggests the modified conjecture that 1 ≤ sdγ(G) ≤ 4 for any graph G. It is also conjectured in  that for every graph G with minimum degree δ(G) ≥ 2, sdγ(G) ≤ δ(G) + 1. In this paper we extend several previous results and consider evidence in support of these two conjectures.
Favaron, Odile; Haynes, Teresa W.; and Hedetniemi, Stephen T.. 2004. Domination Subdivision Numbers in Graphs. Utilitas Mathematica. Vol.66 195-209. https://doi.org/10.7151/dmgt.1147 ISSN: 0315-3681