Throttling zero force propagation speed on graphs

Zero forcing is a game played on a graph that starts with a coloring of the vertices as white and black and at each step any vertex colored black with a unique neighbor colored white “forces” the color of the white vertex to become black. In this note we look at what happens when we balance the size of the initial set of vertices colored black and the number of steps, called speed of propagaion, that it takes for all vertices to be colored black. We also give an example that shows it is possible in some graphs to slow down the speed of propagation in the graph by choosing larger initial sets. Finally, we give a tight relationship between the zero forcing number and the number of edges in the graph.