connected zero forcing sets and connected propagation time of graphs

The zero forcing number z(g) of a graph g is the minimum cardinality of a set s with colored (black) vertices which forces the set v (g) to be colored (black) after some times. “color change rule”: a white vertex is changed to a black vertex when it is the only white neighbor of a black vertex. in this case, we say that the black vertex forces the white vertex. we investigate here the concept of connected zero forcing set and connected zero forcing number. we discusses this subject for special graphs and some products of graphs. also we introduce the connected propagation time. graphs with extreme minimum connected propagation times and maximum propagation times |g| − 1 and |g| − 2 are characterized.