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 S. The domination number gamma(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number 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 June 2000, Arumugam [1] conjectured that 1 less than or equal to sd(gamma)(G) less than or equal to 3 for any graph G. However, a counterexample to this conjecture given in [6] suggests the modified conjecture that 1 less than or equal to sd(gamma)(G) less than or equal to 4 for any graph G. It is also conjectured in [6] that for every graph G with minimum degree delta(G) greater than or equal to 2, sd(gamma)(G) less than or equal to delta(G) + 1. In this paper we extend several previous results and consider evidence in support of these two conjectures.