Leaky Zero Forcing on Induced Subgraphs of $d$-dimensional Grid Graphs with an Application to Hopi Rectangles

We study zero forcing and $\mathrm{\ell }$-leaky zero forcing on induced subgraphs of $d$-dimensional grid graphs. Using $\mathrm{\ell }$-leaky forts, we prove structural results showing that for $\mathrm{\ell }\le 2d-1$, every nonempty $\mathrm{\ell }$-leaky fort in an induced subgraph of $P_{n_1}\mathrm{\square }\cdots \mathrm{\square }{P}_{n_d}$ intersects the boundary of the graph. These results give general bounds and, in certain settings, exact values for the $\mathrm{\ell }$-leaky forcing number of induced subgraphs. Motivated by this framework, we introduce an integer lattice based definition of the Hopi rectangle graphs $HD(a,b)$ as induced subgraphs of $P_{a+b}\mathrm{\square }{P}_{a+b}$. For this particular family of graphs, we show that the zero forcing number equals the maximum nullity, and we completely characterize the $\mathrm{\ell }$-leaky forcing number for all $\mathrm{\ell }\ge 1$.