Let G be a simple graph. A set   is a connected super dominating set if for every  there exists an external private neighbor of v with respect to S and the subgraph  induced by S is connected. The connected super domination number of G, denoted by  is the minimum cardinality of a connected super dominating set in G. A connected super dominating set S of G with  is called a -set of G.

In this paper, the connected super dominating sets of some common graphs and the graphs resulting from the join, corona, lexicographic product and Cartesian product of graphs are characterized. Also, the connected super domination numbers of these graphs are determined.

domination, super domination, connected super domination, join, corona, lexicographic product, Cartesian product.