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

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 $\ell\ge 1$.

💡 Research Summary

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The paper investigates the interplay between zero forcing, a graph‑coloring process, and its fault‑tolerant variant called ℓ‑leaky zero forcing, focusing on induced subgraphs of d‑dimensional grid graphs and a particular family of planar graphs known as Hopi rectangle graphs (also called Aztec rectangle graphs).

Zero forcing begins with a set of blue vertices; a blue vertex with exactly one white neighbor forces that neighbor to become blue. The minimum size of an initial blue set that eventually turns the whole graph blue is the zero forcing number Z(G). This combinatorial parameter is known to bound the maximum nullity M(G) of the family of real symmetric matrices whose off‑diagonal zero pattern matches the graph: M(G) ≤ Z(G).

In many applications (e.g., quantum control, sensor networks) some vertices may be faulty and unable to perform forces. The ℓ‑leaky zero forcing model captures this by allowing up to ℓ vertices to be designated as “leaks”. The ℓ‑leaky forcing number Z^{(ℓ)}(G) is the smallest size of an initial blue set that can force the entire graph blue regardless of which ℓ vertices are leaked.

A central tool is the notion of an ℓ‑leaky fort: a vertex set S such that at most ℓ vertices outside S have exactly one neighbor inside S. Proposition 6 (from earlier work) shows that a set B is an ℓ‑leaky forcing set iff B meets every ℓ‑leaky fort. Thus, characterizing forts yields bounds on Z^{(ℓ)}.

The authors first study arbitrary induced subgraphs H of the d‑dimensional Cartesian product P_{n₁}□…□P_{n_d}. Lemma 7 proves a geometric fact: if R′⊆V(H) consists of vertices of full degree 2d (the interior of the grid), then there exist at least 2d distinct vertices outside R′ each having exactly one neighbor in R′. Using this, Theorem 8 shows that for any ℓ ≤ 2d − 1, every non‑empty ℓ‑leaky fort must contain a vertex of degree ≤ 2d − 1; in other words, any such fort intersects the boundary δH = {v∈V(H) : deg_H(v) ≤ 2d − 1}.

From Theorem 8 the authors derive immediate corollaries. Corollary 9 gives the two‑sided inequality
 |S_ℓ| ≤ Z^{(ℓ)}(H) ≤ |δH|,
where S_ℓ = {v : deg_H(v) ≤ ℓ}. When ℓ = 2d − 1 the lower and upper bounds coincide, yielding the exact value Z^{(2d‑1)}(H) = |δH|. Corollary 10 further states that if every vertex has degree either ≤ ℓ or exactly 2d (i.e., the graph consists of a full interior and a low‑degree boundary), then Z^{(ℓ)}(H) = |S_ℓ|. These results collapse the general bounds to closed formulas for many natural grid‑induced families.

The second part of the paper applies the theory to Hopi rectangle graphs HD(a,b). These are defined as induced subgraphs of the square grid P_{a+b}□P_{a+b} that form a “staircase” shape parameterized by integers a,b ≥ 1. Historically they arise as the duals of Aztec diamond graphs and have been studied in the context of domino tilings. The authors first prove that for HD(a,b) the zero forcing number equals the maximum nullity, extending known results for Aztec diamonds. This is done by constructing explicit zero forcing sets that meet the lower bound M(HD(a,b)) and by showing that any symmetric matrix respecting the graph’s adjacency pattern attains that nullity.

Next, they determine Z^{(ℓ)}(HD(a,b)) for all ℓ ≥ 1. For ℓ ≤ 3 (recall d = 2, so 2d‑1 = 3) the previous corollaries apply directly: every ℓ‑leaky fort must intersect the boundary, and the boundary consists of the vertices on the four outer edges of the rectangle, whose count is 2(a+b) − 4. Hence Z^{(ℓ)}(HD(a,b)) = 2(a+b) − 4 for ℓ = 1,2,3.

When ℓ > 3, Lemma 7 guarantees the existence of four distinct “outside” vertices each with a unique neighbor in any interior set of full‑degree vertices. Consequently a fort can avoid the boundary only if it contains enough interior vertices to absorb more than ℓ such unique‑neighbor vertices. By carefully counting these configurations, the authors derive an exact formula:

 Z^{(ℓ)}(HD(a,b)) = 2(a+b) − 4 + (ℓ − 3) = a + b + ℓ − 3.

In other words, each additional leak beyond the critical value 3 forces the initial blue set to grow by one vertex. The proof combines the structural lemmas with explicit constructions of ℓ‑leaky forcing sets that meet the lower bound, showing optimality.

The paper also includes a brief cultural note: the term “Hopi rectangle” was chosen for euphonic reasons rather than strict ethnographic accuracy, acknowledging that the underlying motif appears across several pre‑Columbian cultures.

In conclusion, the work makes three substantive contributions:

1. It establishes a general structural theorem for ℓ‑leaky forts in induced subgraphs of d‑dimensional grids, proving that for ℓ ≤ 2d‑1 every fort must touch the low‑degree boundary.
2. It translates this structural insight into tight bounds—and in many cases exact values—for the ℓ‑leaky forcing number of such subgraphs, unifying and extending several earlier results.
3. It applies the theory to Hopi rectangle graphs, proving that their zero forcing number equals the maximum nullity and providing a complete, closed‑form description of Z^{(ℓ)} for all ℓ.

These results deepen the connection between combinatorial forcing processes, linear algebraic graph invariants, and the geometry of lattice graphs, and they suggest further avenues such as extending the analysis to non‑rectangular lattice regions, higher‑dimensional analogues of Aztec structures, or dynamic leak models where the set of faulty vertices may change over time.