Isoperimetric Inequalities for Polyominoes

• The paper establishes that the minimal perimeter of polyominoes is characterized by the Harary–Harborth result and extends this framework to include holes and nonlocal perimeters.
• It derives topological bounds connecting the number of holes to the overall structure using dual graph techniques and edge counting methods.
• The work introduces nonlocal bi-axial perimeter functionals, linking discrete geometric minimizers to metastability phenomena in statistical physics and suggesting paths for higher-dimensional studies.

A polyomino is a finite, edge-connected union of unit squares in the plane, serving as a discrete analog of measurable planar domains. Isoperimetric inequalities for polyominoes rigorously constrain the interplay between area, perimeter, topology (notably holes), and combinatorial structure, quantifying extremal configurations that minimize or maximize key quantities such as the perimeter given area and topological complexity. Recent advances extend this classical framework by incorporating nonlocal perimeter notions, connecting the combinatorics of polyominoes to statistical physics and discrete geometry.

1. Classical Isoperimetric Inequality for Polyominoes

The perimeter of a polyomino $A\subset\mathbb{R}^2$ consisting of $n$ unit squares (“tiles”) is denoted $p(A)$. The minimal perimeter achievable with $n$ tiles, denoted $p_{\text{min}}(n)$, is classically given by the Harary–Harborth result:

$p_{\text{min}}(n) = 2\lceil 2\sqrt{n} \rceil.$

This minimum is realized by squares or quasi-squares and a small number of related shapes. For $n=l^2$, the unique minimizer is the $l\times l$ square (Malen et al., 2019).

2. Topological Isoperimetric Inequality and the Role of Holes

Polyominoes may possess $h$ holes, i.e., bounded components of $\mathbb{R}^2 \setminus A$. The topological boundary decomposes into the outer boundary $p_o(A)$ and total hole boundary $p_h(A)$. Each hole has at least 4 boundary edges, $p_h(A) \geq 4h$. Let the dual graph have $b(A)\geq n-1$ edges (with equality exactly for acyclic arrangements).

The “topological isoperimetric inequality” constrains $h$ as a function of $n$:

$$h \leq M(n,h) := \frac{2n+2 - p_{\text{min}}(n+h)}{4}.$$

This is derived via edge counting and a hole-filling argument ensuring $p_o(A)\geq p_{\text{min}}(n+h)$. The critical case of equality yields structural restrictions on the dual graph and the spatial allocation of holes (Malen et al., 2019).

3. Extremal Problems and Crystallized Polyominoes

The extremal problem seeks $g(h)$, the minimum number of tiles required to form a polyomino with $h$ holes:

$$g(h) = \min\{ n: \text{ there exists } A \text{ with } |A|=n, h(A)=h \}.$$

Efficient (attaining equality in the inequality above) and, in almost all cases, unique constructions called crystallized polyominoes realize this optimum. Explicit recurrence and piecewise closed-form expressions for $g(h)$ are known, with combinatorial benchmarks at $n+h$ equal to perfect squares or pronic numbers. Asymptotically, $g(h) = 2h + O(\sqrt{h})$, with machine enumeration providing $g(h)$ for small $h$ (Malen et al., 2019).

Key features of crystallized polyominoes:

• Dual graph is a tree ($b(A) = n - 1$).
• All holes have area 1 tile.
• $p_o(A) = p_{\text{min}}(n + h)$.
• Examples include the ring of 7 (for $h=1$, $g(1)=7$) and the unique $S_l$ family for $h_l = (2^{2l}-1)/3$.

4. Nonlocal Isoperimetric Inequalities and Bi-Axial Perimeter

Recent developments generalize the perimeter concept to nonlocal, bi-axial discrete perimeters. For $\lambda>1$, define for a polyomino $\mathcal P$:

$$P_\lambda(\mathcal{P}) = \sum_{\substack{x \in \mathbb{Z}^2 \cap \mathcal{P} \ y \in \mathbb{Z}^{2} \setminus \mathcal{P}}} \left( \mathbf{1}_{x_2 = y_2} \frac{1}{|x_1 - y_1|^{\lambda}} + \mathbf{1}_{x_1 = y_1} \frac{1}{|x_2 - y_2|^{\lambda}} \right).$$

Here, all internal and external row/column interactions between boundary pairs contribute. For $\lambda \to \infty$, $P_\lambda$ converges to the standard perimeter.

The nonlocal discrete isoperimetric inequality asserts that for all $n\in\mathbb{N}$ and $\lambda$ above a critical value $\lambda_0 \approx 1.8$, the minimizers of $P_\lambda$ for area $n$ polyominoes are always among explicit classes: squares, quasi-squares, rectangles with protuberances, and structurally constrained variants. For each $n$, one computes these minimizers via explicit Hurwitz–ζ-based formulae. The result extends classical isoperimetry to functionals with long-range dependencies and provides a precise minimizer classification (Jacquier et al., 2024).

5. Classification and Properties of Minimizers

The minimizer classification under both local and nonlocal perimeters is structurally robust. For all large enough $\lambda$, minimizers are:

• Perfect squares ($n=l^2$),
• Quasi-squares ($n=l(l+1)$),
• Rectangles or quasi-squares with single protuberances,
• In specific cases, rectangles sharing equal classical perimeter.

Closed-form expressions for $P_\lambda$ are available in terms of Hurwitz–ζ functions, with uniqueness up to symmetries and, in degenerate cases, tied minimal values for shapes sharing the same perimeter (Jacquier et al., 2024).

These minimizers are characterized by cross-convexity: stripwise, they cannot be further compacted to lower $P_\lambda$, and nonconvex or disconnected arrangements are never optimal.

6. Connections to Statistical Physics and Generalizations

The nonlocal bi-axial perimeter arises naturally in the study of metastability for two-dimensional Ising models with long-range interactions. In this context, the minimal-energy configurations of plus-spin droplets correspond precisely to the isoperimetric minimizers for $P_\lambda$. The identification of critical droplets and energy barriers relies on this combinatorial geometric minimization, with the critical size growing dramatically as $\lambda \searrow 1$ (Jacquier et al., 2024).

Beyond the square lattice, analogous extremal and isoperimetric problems are posed for 3D polycubes and for discrete domains in other lattices, with homological and geometric-topological bounds. Enumeration of all free crystallized polyominoes for large $h$ remains open, with expansion–compression dynamical constructions suggesting possible approaches, but boundary-move variants introduce combinatorial complexity (Malen et al., 2019).

7. Open Problems and Research Directions

Major unresolved questions include:

• Enumeration and classification of all free crystallized polyominoes beyond small $h$.
• Detailed analysis of exceptional parameter sets where unique efficiently structured minimizers do not exist.
• Generalization of isoperimetric inequalities to polycubes and higher-dimensional discrete complexes.
• Further exploration of nonlocal isoperimetric functionals, including those modeling different physical systems or modified interaction rules.

These directions continuously bridge combinatorial, geometric, and statistical-physical approaches, establishing isoperimetric theory for polyominoes as a nexus for discrete geometry, algebraic topology, and mathematical physics (Malen et al., 2019, Jacquier et al., 2024).