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The zero forcing number $Z(G)$ of a graph $G$ is the minimum cardinality of aset $S$ 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 ablack vertex when it is the only white neighbor of a black vertex. In thiscase, we say that the black vertex forces the white vertex. We investigate herethe concept of connected zero forcing set and connected zero forcing number. Wediscusses this subject for special graphs and some products of graphs. Also weintroduce the connected propagation time. Graphs with extreme minimum connectedpropagation times and maximum propagation times $|G|-1$ and $|G|-2$ arecharacterized.

Connected zero forcing sets and connected propagation time of graphs