Connected domination value in graphs

In a connected graph G = ( V,E ), a set D ⊂ V is a connected dominating set if for every vertex v ∈ V \ D , there exists u ∈ D such that u and v are adjacent, and the subgraph〈 D 〉induced by D in G is connected. A connected dominating set of minimum cardinality is called a γ c -set of G . For each vertex v ∈ V , we define the connected domination value of v to be the number of γ c -sets of G to which v belongs. In this paper, we study the properties of connected domination value of a connected graph G and its relation to other parameters of a connected graph. Finally, we compute the connected domination value and number of γ c -sets for a few well-known family of graphs.