Failed 2-Distance Zero Forcing in Graphs

Let G be a graph with vertex and edge-sets $E(G)$ and $V(G)$, respectively. Then M⊂ V(G) is called a failed 2-distance zero forcing set of G if M is not a 2-distance zero forcing set of G. The failed 2-distance zero forcing number of a graph G, denoted by F²(G), is the maximum cardinality of a failed 2-distance zero forcing set. In this paper, we introduce the said parameter and study this on some graphs and on the join of two graphs. Moreover, we define the failed co-zero forcing concept to characterize the failed 2-distance zero forcing sets in the join of two graphs, and finally derive the formulas for solving the exact values of the parameter.