Some necessary conditions for graphs with extremal connected 2-domination number

Let G be a graph with no multiple edges and loops. A subset S of the vertex set of G is a dominating set ofhas at least k neighbors in S and the subgraph GS is connected. The domination number of G is the number of vertices in a minimum dominating set of G, denoted by γ (G). The connected k-domination number of G, denoted by γ c k (G), is the minimum cardinality of a connected k-dominating set of G. For k = 1, we simply write γc (G). It is known that the bounds γ c 2 (G) γ (G) + 1 and γ c 2 (G) γc (G) + 1 are sharp. In this research article, we present the necessary condition of the connected graphs G with γ c 2 (G) = γ (G) + 1 and the necessary condition of the connected graphs G with γ c 2 (G) = γc (G) + 1. Moreover, we present a graph construction that takes in any connected graph with r vertices and gives a graph G with γ c 2 (G) = r, γc (G) = r -1, and γ (G) ∈ r -1, r -2.