Disproof of a Conjecture on the Subdivision Domination Number of a Graph

A set S of vertices of a graph G = (V,E) is a dominating set if every vertex of $$V(G) \setminus S$$ is adjacent to some vertex in S. The domination number γ(G) is the minimum cardinality of a dominating set of G. 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. Haynes et al. (Discussiones Mathematicae Graph Theory 21 (2001) 239-253) conjectured that $${\rm sd}_{\gamma} (G) \le \delta(G) + 1$$ for any graph G with $$\delta(G) \ge 2$$. In this note we first give a counterexample to this conjecture in general and then we prove it for a particular class of graphs.