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

TL;DR

This work analyzes zero forcing and $\ell$-leaky zero forcing on induced subgraphs of $d$-dimensional grid graphs, introducing leaky forts as a structural tool. Focusing on Hopi rectangle graphs $HD(a,b)$, induced in $P_{a+b}\square P_{a+b}$, it establishes that $Z(HD(a,b))=M(HD(a,b))=a+b$ and completely characterizes $Z_{(\ell)}(HD(a,b))$ for all $\ell\ge1$: $Z_{(1)}=a+b$, $Z_{(2)}=Z_{(3)}=2(a+b)$, and $Z_{(\ell)}=|V(HD(a,b))|=a+b+2ab$ for $\ell\ge4$. The results leverage boundary- and degree-structure arguments, with a general framework applicable to broader grid-induced subgraphs and potential extensions to higher dimensions and polyomino families. The findings connect zero forcing to maximum nullity in this family and provide exact leaky forcing spectra that illuminate network resilience under leaks.

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 $\ell\ge 1$.

Figures (5)

• Figure 1:
• Figure 2:
• Figure 3: Initial steps of zero forcing on $HD(2,3)$
• Figure 4: Ordering of forces given by forcing process $F_2$
• Figure 5: Ordering of forces given by forcing process $F_3$

Theorems & Definitions (31)

• Proposition 1: Proposition 2.4, AIMMINIMUMRANKWorkgroup
• Lemma 2: Lemma 2.1, KenterOGpaper
• Lemma 3: Lemma 2.4, KenterOGpaper
• Theorem 4: Theorem 2.3, Reslience
• Definition 5
• Proposition 6: Proposition 2.3, KenterOGpaper
• Lemma 7
• proof
• Theorem 8
• proof