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 gamma(G). The connected k-domination number of G, denoted by gamma(c)(k) (G), is the minimum cardinality of a connected k-dominating set of G. For k = 1, we simply write gamma(c)(G). It is known that the bounds gamma(c)(2)(G) >= gamma(G) + 1 and gamma(c)(2)(G) >= gamma(c)(G) + 1 are sharp. In this research article, we present the necessary condition of the connected graphs G with gamma(c)(2)(G) = gamma(G) + 1 and the necessary condition of the connected graphs G with gamma(c)(2)(G) = gamma(c)(G)+1. Moreover, we present a graph construction that takes in any connected graph with r vertices and gives a graph G with gamma(c)(2)(G) = r, gamma(c)(G) = r - 1, and gamma(G) is an element of {r - 1, r - 2}.