Asian Research Journal of Mathematics

A subset Z_k of vertices in a graph is called a k-forcing set if its vertices are initially colored black, while all remaining vertices are uncolored. The graph then undergoes a color propagation process governed by the following rule: a black vertex with at most k uncolored neighbors forces each of those neighbors to become black. This process continues until all vertices in the graph are colored black. The k-forcing number of a graph, denoted as Z_k(G), represents the smallest possible size of a k-forcing set. The propagation time of a k-forcing set of graph G is the minimum number of steps that it takes to force all the vertices black, starting with the vertices in the k-forcing set and performing independent k-forces simultaneously. The minimum and maximum k-forcing propagation times of a graph are taken over all minimum k-forcing sets of the graph. We discuss the minimum k-forcing propagation time of graph G. Additionally, it delves into the precise determination of bounds for k-forcing propagation times. for certain well-known graphs. Also characterized regular graphs with k-forcing propagation time one.