Connected D-centro dominating sets in graphs

Let G = (V, E) be a connected graph. A subset S ⊂ V(G) of a graph G is said to be D-centro dominating set of G if for every v ∈ V − S, there exists a vertex u in S such that D(u, v) = Rad(G). The minimum cardinality of the D-centro dominating set is called D-centro domination number, denoted by DCγ(G). A connected D-centro dominating set of G is an D-centro dominating set S such that the subgraph < S > induced by S is connected. The minimum cardinality of a connected D-centro dominating set of G is the connected D-centro domination number of G and it is denoted by cDCγ(G). A connected D-centro dominating set of cardinality cDCγ(G) is called a cDCγ-set of G. Some bounds for the connected D-centro domination number are determined. An important realization result on connected D-centro domination number is proved that for any integers a, b with 3 < a ≤ b, there exists a connected graph G such that DCγ(G) = a, and cDCγ(G) = b.