Zero Forcing, Leaky Zero Forcing, and Leaky Forts on Hopi Rectangle Graphs

Abstract: We study the family of Hopi rectangle graphs, a planar generalization of the well-studied Aztec diamond graphs. We provide a new definition of this graph family as induced subgraphs of the Cartesian product of two path graphs, which helps frame our study. We then establish that the maximum nullity and the zero forcing number coincide for all Hopi rectangle graphs. Extending to the fault tolerant framework of leaky zero forcing, we use the notion of leaky forts to get a complete characterization of the $\ell$-leaky forcing number for all $\ell\ge 1$ in Hopi rectangle graphs.