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arXiv · 2509.21529

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

Ryan Moruzzi Jr·Sagar Shah·Aaditeya Tripathi
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Abstract

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

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Ryan Moruzzi Jr, Sagar Shah, Aaditeya Tripathi. 2026-02-12. Leaky Zero Forcing on Induced Subgraphs of $d$-dimensional Grid Graphs with an Application to Hopi Rectangles. https://arxiv.org/abs/2509.21529

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