All Graphs with a Failed Zero Forcing Number of Two

Given a graph G, the zero forcing number of G, Z(G), is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Zero forcing numbers have attracted great interest over the past 15 years and have been well studied. In this paper, we investigate the largest size of a set S that does not force all of the vertices in a graph to be in S. This quantity is known as the failed zero forcing number of a graph and will be denoted by F(G). We present new results involving this parameter. In particular, we completely characterize all graphs G where F(G)=2, solving a problem posed in 2015 by Fetcie, Jacob, and Saavedra.

Zero forcing number of fuzzy graphs with application

We introduce and study forcing number for fuzzy graphs. Also, we compute zero forcing numbers for some classes of graphs and extend this concept to fuzzy graphs. In this regard we obtain upper bounds for zero forcing of some classes of fuzzy graphs. We will proceed to obtain a new algorithm to computing zero forcing set and finding a formula for zero forcing number, and by some examples we illustrate these notions. Finally, we introduce some applications of fuzzy zero forcing in medical treatments.