On the Semitotal Forcing Number of a Graph

Zero forcing is an iterative graph coloring process that starts with a subset S of "colored" vertices, all other vertices being "uncolored". At each step, a colored vertex with a unique uncolored neighbor forces that neighbor to be colored. If at the end of the forcing process all the vertices of the graph are colored, then the initial set S is called a zero forcing set. If in addition, every vertex in S is within distance 2 of another vertex of S, then S is a semitotal forcing set. The semitotal forcing number F-t2(G) of a graph G is the cardinality of the smallest semitotal forcing set of G. In this paper, we begin to study basic properties of F-t2(G), relate F-t2(G) to other domination parameters, and establish bounds on the effects of edge operations on the semitotal forcing number. We also investigate the semitotal forcing number for subfamilies of cubic graphs.