In a graph G(V, E), a set d V(G) is said to be a monopoly set of G if any vertex v V - D has at least neighbors in D, where is a degree of in . A monopoly set of is called a connected monopoly set of if the subgraph induced by is connected. The minimum cardinalities of connected monopolies sets of , denoted by is called the connected monopoly size of . In this paper, we investigate the relationship between and some other parameters of graphs. Bounds for and its exact values for some standard graphs are found.