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 of G if every vertex in V ( G ) \ S is adjacent to at least one vertex of S . A connected k -dominating set of G is a subset S of the vertex set V ( G ) such that every vertex in V ( G ) \ S has at least k neighbors in S and the subgraph G [ S ] 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 ck ( G ) , is the minimum cardinality of a connected k -dominating set of G . For k = 1 , we simply write γ c ( G ) . It is known that the bounds γ c 2 ( G ) (cid:62) γ ( G ) + 1 and γ c 2 ( G ) (cid:62) γ c ( 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 ) = γ c ( 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 , γ c ( G ) = r − 1 , and γ ( G ) ∈ { r − 1 , r −