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.