On the zero forcing number of the complement of graphs with forbidden subgraphs

Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on n vertices is either $n-3$ or $n-1$ in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.