Hopi Rectangle Graphs Overview

• Hopi rectangle graphs are planar graphs defined as induced subgraphs of Cartesian products of paths, generalizing Hopi diamond graphs with a lattice-based formulation.
• They are characterized by a closed-form vertex count |HD(m,n)| = m+n+2mn and an edge covering by mn four-cycles, which facilitates precise rank and nullity analysis.
• These graphs exhibit an exact equality between the zero forcing number and maximum nullity, while adjustable leaky forcing numbers reveal nuanced fault-tolerance properties.

Searching arXiv for the primary paper and closely related terminology to ground the article. Searching for (Jr et al., 25 Sep 2025) and related Hopi/Aztec rectangle graph terminology. Hopi rectangle graphs are a planar family of graphs that generalize Hopi (Aztec) diamond graphs. For parameters $m,n \in \mathbb{N}$, the Hopi rectangle graph of order $(m,n)$, denoted $HD(m,n)$, is realized as an induced subgraph of a Cartesian product of paths, and this lattice-based formulation supports a complete determination of its zero forcing number, maximum nullity, and $\ell$-leaky forcing number for every $\ell \ge 1$ (Jr et al., 25 Sep 2025).

1. Lattice definition and geometric structure

For graphs $G_1$ and $G_2$, the Cartesian product $G_1 \square G_2$ has vertex set $V(G_1 \square G_2)=V(G_1)\times V(G_2)$, with

$(x_1,y_1)\sim (x_2,y_2)$

if and only if either $(m,n)$0 and $(m,n)$1 in $(m,n)$2, or $(m,n)$3 and $(m,n)$4 in $(m,n)$5. Writing $(m,n)$6 for the path on $(m,n)$7 vertices, typically with $(m,n)$8 and $(m,n)$9 when $HD(m,n)$0, the Hopi rectangle graph of order $HD(m,n)$1 is the induced subgraph of $HD(m,n)$2 on

$HD(m,n)$3

Its edges are inherited from the ambient grid: $HD(m,n)$4

This description may be viewed as starting from the $HD(m,n)$5 grid and removing four corner regions by the boundary inequalities $HD(m,n)$6, $HD(m,n)$7, and $HD(m,n)$8. Under this indexing convention, the vertex with largest $HD(m,n)$9-coordinate is $\ell$0. The family is symmetric in the sense that

$\ell$1

That symmetry is structural rather than merely numerical: the same parameter formulas and forcing phenomena hold after exchanging the two side lengths (Jr et al., 25 Sep 2025).

2. Relation to Hopi diamonds and basic enumerative formulas

Earlier formulations define the Hopi diamond graph of order $\ell$2 as a subgraph of the $\ell$3-grid with diagonal adjacencies and parity restriction

$\ell$4

A visual formulation places vertices on unit squares of a $\ell$5 chessboard and joins diagonally adjacent squares, producing two components: the even Hopi rectangle $\ell$6 and the odd $\ell$7. When $\ell$8, the even Hopi rectangle becomes the Hopi diamond graph. The lattice definition above coincides, after a $\ell$9 rotation about $\ell \ge 1$0, with the even Hopi rectangle $\ell \ge 1$1, and when $\ell \ge 1$2 it recovers the Hopi diamond graphs.

The order of the graph admits a closed form. The base case is

$\ell \ge 1$3

and in general

$\ell \ge 1$4

This formula supplies the ambient scale for all later parameter identities. It also interacts cleanly with the family’s cycle structure: $\ell \ge 1$5 admits an edge covering by $\ell \ge 1$6 copies of $\ell \ge 1$7,

$\ell \ge 1$8

a fact that is decisive in the minimum-rank and maximum-nullity analysis (Jr et al., 25 Sep 2025).

3. Zero forcing and the equality $\ell \ge 1$9

The zero forcing process begins with an initial blue set $G_1$0 and repeatedly applies the color change rule: if a blue vertex $G_1$1 has exactly one white neighbor $G_1$2, then $G_1$3 forces $G_1$4, written $G_1$5, and $G_1$6 becomes blue. The minimum size of a set forcing all vertices blue is the zero forcing number $G_1$7. Its linear-algebraic significance comes from the standard inequality

$G_1$8

where $G_1$9 is the maximum nullity over real symmetric matrices whose off-diagonal support is the graph.

For Hopi rectangle graphs, the edge covering by $G_2$0 four-cycles gives

$G_2$1

because $G_2$2 and $G_2$3 for an edge covering by subgraphs $G_2$4. Combining this with $G_2$5 yields

$G_2$6

A matching upper bound for zero forcing is obtained from the explicit boundary set

$G_2$7

This set consists of two disjoint boundary lines, one on the lower-left and one on the upper-left, and satisfies $G_2$8. It is a zero forcing set for $G_2$9: in the first round, each vertex on $G_1 \square G_2$0 forces its unique white neighbor to the right, each vertex on $G_1 \square G_2$1 does the same, and subsequent rounds continue horizontal forces along each row until the graph is entirely blue.

The resulting identity is exact: $G_1 \square G_2$2 The proof is a sharp squeezing argument: the lower bound $G_1 \square G_2$3, the constructive upper bound $G_1 \square G_2$4, and the general inequality $G_1 \square G_2$5 force equality throughout. In this family, the linear-algebraic parameter $G_1 \square G_2$6 and the combinatorial parameter $G_1 \square G_2$7 therefore coincide for all orders $G_1 \square G_2$8 (Jr et al., 25 Sep 2025).

4. Leaky zero forcing and leaky forts

Leaky zero forcing introduces faults. Up to $G_1 \square G_2$9 vertices are designated as leaks, meaning they cannot force. The $V(G_1 \square G_2)=V(G_1)\times V(G_2)$0-leaky forcing number $V(G_1 \square G_2)=V(G_1)\times V(G_2)$1 is the minimum size of an initial set that forces all vertices blue for every placement of $V(G_1 \square G_2)=V(G_1)\times V(G_2)$2 leaks. Two general facts govern the analysis: $V(G_1 \square G_2)=V(G_1)\times V(G_2)$3 and every $V(G_1 \square G_2)=V(G_1)\times V(G_2)$4-leaky forcing set contains all vertices of degree at most $V(G_1 \square G_2)=V(G_1)\times V(G_2)$5.

The relevant obstruction is an $V(G_1 \square G_2)=V(G_1)\times V(G_2)$6-leaky fort. A set $V(G_1 \square G_2)=V(G_1)\times V(G_2)$7 is an $V(G_1 \square G_2)=V(G_1)\times V(G_2)$8-leaky fort if:

1. for all $V(G_1 \square G_2)=V(G_1)\times V(G_2)$9, either $(x_1,y_1)\sim (x_2,y_2)$0 or $(x_1,y_1)\sim (x_2,y_2)$1; and
2. $(x_1,y_1)\sim (x_2,y_2)$2.

The duality statement is exact: $(x_1,y_1)\sim (x_2,y_2)$3 is an $(x_1,y_1)\sim (x_2,y_2)$4-leaky forcing set if and only if $(x_1,y_1)\sim (x_2,y_2)$5 intersects every $(x_1,y_1)\sim (x_2,y_2)$6-leaky fort.

For $(x_1,y_1)\sim (x_2,y_2)$7, the case $(x_1,y_1)\sim (x_2,y_2)$8 is resolved by reusing the same boundary set $(x_1,y_1)\sim (x_2,y_2)$9. The proof constructs three chronological forcing processes $(m,n)$00 from $(m,n)$01: $(m,n)$02 uses horizontal forces to the right row-by-row, $(m,n)$03 alternates left-to-right and bottom-to-top forces indexed by diagonals, and $(m,n)$04 alternates left-to-right and top-to-bottom forces indexed by diagonals. Together with the two-way forcing proposition—$(m,n)$05 is a $(m,n)$06-leaky forcing set if and only if every $(m,n)$07 can be forced by two distinct parents—this gives

$(m,n)$08

For higher leak levels, the degree structure becomes decisive. $(m,n)$09 has exactly $(m,n)$10 vertices of degree $(m,n)$11, so every $(m,n)$12-leaky forcing set must contain all of them, yielding

$(m,n)$13

A complementary fort argument uses the degree-$(m,n)$14 core: if $(m,n)$15 is the set of degree-$(m,n)$16 vertices and $(m,n)$17 is nonempty, then there exist at least four vertices outside $(m,n)$18 each having exactly one neighbor in $(m,n)$19. Consequently, every $(m,n)$20-leaky fort in $(m,n)$21 for $(m,n)$22 must contain at least one degree-$(m,n)$23 vertex. Letting $(m,n)$24 be the set of all degree-$(m,n)$25 vertices, one obtains a $(m,n)$26-leaky forcing set of size $(m,n)$27, and monotonicity then forces

$(m,n)$28

For $(m,n)$29, the maximum degree is $(m,n)$30. The low-degree lemma then implies that the only $(m,n)$31-leaky forcing set is the entire vertex set: $(m,n)$32

Parameter	Value on $(m,n)$33
$(m,n)$34	$(m,n)$35
$(m,n)$36	$(m,n)$37
$(m,n)$38	$(m,n)$39
$(m,n)$40	$(m,n)$41
$(m,n)$42	$(m,n)$43
$(m,n)$44, $(m,n)$45	$(m,n)$46

These formulas provide a complete characterization of leaky zero forcing on the family (Jr et al., 25 Sep 2025).

5. Constructive methods and representative examples

The forcing constructions are explicit. A minimum zero forcing set is always

$(m,n)$47

and the propagation can be carried out by scanning rows left-to-right, with newly colored vertices forcing their unique white right neighbors. The propagation requires time linear in $(m,n)$48 under straightforward implementations. For $(m,n)$49, the same set $(m,n)$50 is used, together with one of the chronological processes $(m,n)$51, $(m,n)$52, or $(m,n)$53, or an interleaving of them, to guarantee two distinct potential forcing parents for each uncolored vertex. For $(m,n)$54 or $(m,n)$55, one takes

$(m,n)$56

which has size $(m,n)$57. For $(m,n)$58, the unique $(m,n)$59-leaky forcing set is $(m,n)$60 itself.

Two small examples illustrate the formulas. For $(m,n)$61,

$(m,n)$62

A minimum zero forcing set is

$(m,n)$63

of size $(m,n)$64; the first round forces all immediate right neighbors, and subsequent rounds propagate horizontally until all $(m,n)$65 vertices are blue. For $(m,n)$66, a minimum leaky set is the set of all degree-$(m,n)$67 vertices, of size $(m,n)$68.

For $(m,n)$69,

$(m,n)$70

A minimum zero forcing set is

$(m,n)$71

again with horizontal propagation (Jr et al., 25 Sep 2025).

6. Context, significance, and terminological distinctions

Hopi rectangle graphs sit at the intersection of planar graph structure, minimum-rank theory, and propagation processes. Prior work established $(m,n)$72 for Hopi (Aztec) diamond graphs; the extension to the full Hopi rectangle family $(m,n)$73 enlarges that equality to a genuinely two-parameter planar class. The complete characterization

$(m,n)$74

appears to be new for this broader family. A plausible implication is that the boundary-based geometry of $(m,n)$75, together with its tightly controlled degree pattern, makes it unusually amenable to exact resilience calculations.

The term “rectangle graph” has multiple unrelated meanings in the literature, and Hopi rectangle graphs should not be conflated with them. In geometric graph coloring, “rectangle overlap graphs” are overlap graphs of axis-parallel rectangles or, equivalently, intersection graphs of their boundaries; triangle-free instances satisfy $(m,n)$76 (Krawczyk et al., 2013). In spatial contact-graph theory, “Plattenbau graphs” are touching graphs of axis-aligned rectangles in $(m,n)$77, and every planar $(m,n)$78-colorable graph admits such a representation (Felsner et al., 2020). In the NLS literature, Procesi’s “rectangle graph” $(m,n)$79 is a Cayley-graph-based construction encoding resonant interactions among Fourier modes (Procesi, 2024). In enumerative combinatorics, boundary-chord graphs $(m,n)$80 arise from joining marked boundary points of a rectangle by straight segments (Blomberg et al., 2020). None of these are Hopi rectangle graphs in the sense of the family $(m,n)$81.

Within graph theory proper, the principal significance of Hopi rectangle graphs lies in the exact alignment of structural, linear-algebraic, and fault-tolerant forcing parameters on a nontrivial planar family. Their induced-subgraph realization inside $(m,n)$82, the edge covering by $(m,n)$83 copies of $(m,n)$84, and the explicit boundary forcing sets together yield a complete, constructive, and parameterized description of zero forcing and leaky zero forcing on the family (Jr et al., 25 Sep 2025).