Neighborhood Connected 2-Domination Number and Connectivity of Graphs

A subset $S$ of $V$ is called a dominating set in $G$ if every vertex in $V−S$ is adjacent to at least one vertex in $S.$ A set $S ⊆ V$ is called the neighborhood connected 2-dominating set (nc2d-set) of a graph $G$ if every vertex in $V−S$ is adjacent to at least two vertices in $S$ and the induced subgraph $\langle N (S)\rangle$ is connected. The minimum cardinality of a nc2d-set of $G$ is called the neighborhood connected 2-domination number of $G$ and is denoted by $\gamma_{2nc}(G).$ The connectivity $\kappa(G)$ of a graph $G$ is the minimum number of vertices whose removal results in a disconnected or trivial graph. In this paper we find an upper bound for the sum of the neighborhood connected 2-domination number and connectivity of a graph and characterize the corresponding extremal graphs.