An Inverse Approach for Finding Graphs with a Failed Zero Forcing Number of k

For a given graph G, the zero forcing number of Z(G), is smallest cardinality any set S vertices on which repeated applications rule results in all being included S. The as follows: if vertex v S, and exactly one neighbor u not then added to next iteration. failed was introduced by Fetcie, Jacob, Saavedra defined largest fails force graph. In 2021, Gomez et al. proved that there were 15 graphs with two, but their proof complicated, requiring analysis many families. We present an “inverse” approach where we start see have maximum-sized set. This only shorter characterizes particular our proof, characterize |S|=2, meaning considering simple order two induced subgraphs. also has greater potential for characterizing F(G)>2 since general arguments structure are developed this type applicable sets size.