Deep Holes in Polyominoes: Extremal Structures

• Deep holes in polyominoes are fully enclosed voids defined by disjoint simple cycles, ensuring no shared boundaries with the exterior or other holes.
• Construction methods use grid-based frameworks to maximize hole count, achieving asymptotic growth of hₙ ≈ n/3 through careful lattice arrangements.
• Topological and combinatorial analyses, employing Pick’s theorem and isoperimetric estimates, reveal strict constraints on hole density and polyomino structure.

A deep hole in a polyomino is a bounded connected component of the complement of the polyomino in the plane whose boundary loop is disjoint from both the outer boundary of the polyomino and the boundaries of any other internal holes. This notion imposes strict topological and combinatorial separation: a deep hole must be entirely encircled by the polyomino with no shared vertices or edges with other holes or with the external contour. The study of deep holes includes both the extremal maximization question—given an $n$-omino, what is the maximal number $h_n$ of deep holes possible—and the geometric and enumerative structure of polyominoes with such holes. This area sits at the intersection of classical polyomino theory and topological graph theory, with connections to tiling theory, extremal combinatorics, and discrete geometry (Kamenetsky et al., 2014, Baralic et al., 11 Jan 2026).

1. Formal Definitions and Foundational Properties

An $n$-omino is a polyomino consisting of $n$ connected unit squares on the square lattice. For any lattice subset $P\subset\mathbb{Z}^2$ with $|P|=n$ and a connected interior, the collection of bounded connected components of $\mathbb{R}^2 \setminus P$ are called holes. The unique unbounded component is termed the outer boundary.

Deep hole: A hole $H$ of $P$ is a deep hole if its boundary is a simple lattice loop that is disjoint from the outer boundary of $P$ and from the boundaries of all other holes. Consequently, the set of all boundary loops of $P$ forms a disjoint union of simple cycles, each corresponding either to the outer perimeter or to an individual deep hole (Baralic et al., 11 Jan 2026).

Topologically, this excludes “touching” or “nested” holes: no vertex or edge on the boundary of one hole may be shared with the boundary of any other hole or with the exterior. Deep holes are equivalent to voids entirely surrounded by the polyomino, separated by at least one (lattice) layer of tiles from both the outer perimeter and other holes.

2. Construction Techniques and Lower Bounds

Crude and refined constructions are fundamental in maximizing the number of deep holes for given $n$. The most elementary construction establishes that a single deep hole requires at least 8 squares to form a 3 × 3 frame with one central cell removed. Extensions via rectangular lattices allow for the systematic packing of multiple deep holes.

• For any positive integers $a, b \geq 1$, take a $(2a+1) \times (2b+1)$ rectangle, and remove an $a \times b$ grid of interior $1\times 1$ holes. The resulting polyomino has area $n = 3ab + 2a + 2b + 1$ with $h_n \geq ab$ deep holes, leading to $h_n \geq \lfloor (n-3)/5 \rfloor$ for $b=1$. This construction ensures that each hole is well-separated—the minimal enclosing $2 \times 2$ block for each deep hole is strictly embedded inside the ambient polyomino (Baralic et al., 11 Jan 2026).

A refined, parameterized construction defines for each $a \geq 0$, $n = 12a^2 + 20a + 8 + k$ with $0 \leq k < 24a + 32$, a polyomino $A_n$ that is a $(4a+3)\times(4a+3)$ square with the central $(2a+1)\times(2a+1)$ block removed. This base achieves $f_n = (2a+1)^2$ deep holes (for $k=0$). By careful placement of $k$ additional tiles, the number of deep holes increases at a rate of approximately $\lfloor k/3 \rfloor$, without merging or eliminating preexisting deep holes. Thus, for these $n$, $h_n \geq (2a+1)^2 + \lfloor (k-2)/3 \rfloor$ for $k \geq 8$ (Baralic et al., 11 Jan 2026).

3. Extremal and Asymptotic Results

Determining the maximal $h_n$ for fixed $n$ is addressed using both combinatorial and geometric tools, most notably Pick’s theorem.

Lower and Upper Bounds

• Lower Bound: For all $n \geq 8$,

$$h_n > \frac{n}{3} - \frac{16}{9}\sqrt{3n+1} + \frac{65}{9}$$

established via grid-based constructions and elementary algebraic analysis (Baralic et al., 11 Jan 2026).

• Upper Bound: Utilizing generalized Pick's theorem for multiply-connected shapes and isoperimetric inequalities, any $n$-omino with $h_n$ deep holes satisfies

$$h_n < \frac{n}{3} - \frac{1}{18}\sqrt{48\pi n} + O(1)$$

where $O(1)$ is an absolute constant (Baralic et al., 11 Jan 2026).

By combining both bounds, the asymptotics are pinned down to

$h_n = \frac{n}{3} + o(n)$

as $n \to \infty$. That is, for large $n$, the maximum possible number of deep holes in any $n$-omino grows like $n/3$ with sublinear error (Baralic et al., 11 Jan 2026).

Exact Formulas for Infinite Subsequences

For $n = 12a^2 + 20a +8 + k$ ($0\leq k < 24a+32$) and $k$ belonging to clearly specified integer ranges (see (Baralic et al., 11 Jan 2026) for 13 explicit subcases), the construction $A_n$ realizes

$$h_n = (2a+1)^2 + \left\lfloor \frac{k-2}{3} \right\rfloor.$$

In particular, $n = 12a^2 + 20a + 8 \implies h_n = (2a+1)^2$ and $$n = 12a^2 + 32a + 21 \implies h_n = (2a+1)^2 + (4a+3)$$ describe infinite families where the lower and upper bounds coincide (Baralic et al., 11 Jan 2026).

4. Topological and Combinatorial Implications

The requirement that boundaries are disjoint enforces that deep holes are completely encircled by polyomino walls with no adjacency to other holes or the polyomino's exterior. This ensures that holes are "protected" configurations, providing a geometric and topological separation that prohibits degenerate or marginal holes.

A key implication is that achieving many deep holes requires careful management of lattice area. The base case—removing a central square block from a larger square—maximizes interior, disconnected voids, while additional lattice area must be absorbed in such a way as to increment the deep hole count without causing overlap or merger of boundary circuits. A plausible implication is that attempts to nest or closely cluster holes are topologically penalized, reducing the achievable $h_n$ for those configurations.

Isoperimetric analysis shows that hole creation incurs explicit boundary costs. Each deep hole requires an enclosing loop of at least four lattice points (a minimal perimeter), and these perimeters must be non-overlapping. This restricts the density and arrangement of holes within $P$, tightly correlating the count of deep holes to the total available area.

5. Relations to Holey Polyominoes and Tiling Theory

The concept of deep holes generalizes the more familiar context of holey polyominoes—polyominoes with transparent (non-covering) cells introduced specifically to mediate connectivity for tiling and enumeration problems (Kamenetsky et al., 2014). In the context of tiling theory, the minimal number of transparent squares required to connect visible blocks is a measure of “hole depth.” Deep holes, as defined in the extremal context, represent the maximal isolation and enclosure possible for internal voids, as opposed to the minimal connecting requirements of transparent squares.

The study of maximal deep holes is also closely related to the issue of polyomino rectifiability. In tiling, the presence of interior holes—especially those that cannot be filled by the visible squares of any translated copy—renders certain polyominoes unrectifiable (unable to tile a rectangle at all) (Kamenetsky et al., 2014). However, the configurations that maximize deep holes actively avoid perimeters that would permit boundary sharing or filling, instead prioritizing separation and enclosures. This suggests a potential tension between tiling-difficulty and deep-hole maximization.

6. Enumeration, Complexity, and Open Problems

There is no known closed-form generating function enumerating the number of $n$-ominoes with $k$ deep holes, nor a function for the total number of free (up to rotation, reflection, translation) such polyominoes for general $n$ and $k$. The partition of the polyomino's unused lattice area between multiple, mutually isolated holes is combinatorially complex.

Principal open problems include:

• Establishing a complete necessary and sufficient characterization for which $(n,k)$-holey polyominoes are rectifiable for $n>2$ (Kamenetsky et al., 2014).
• Enumerating $n$-ominoes with $k$ deep holes via generating functions $F_{n,k}(x)$.
• Investigating nested (k-tier) holes, leading toward a hierarchy of polyomino “hole-depth.”
• Studying the maximal deep hole count in polyominoes with further constraints or under infinite plane tiling.
• Determining the computational complexity of deciding rectifiability or maximal deep-hole content as $n$ grows (Kamenetsky et al., 2014).

The established results provide both asymptotic and exact formulae for $h_n$ in infinite families of $n$, revealing the tight constraint that $h_n$ cannot grow faster than $n/3 + o(n)$. A plausible implication is that further substantial improvement in $h_n$ for arbitrary $n$ will require fundamentally new geometric or combinatorial constructions.

7. Illustrative Example and Concluding Remarks

For $a=1$, the construction yields $n=12 \cdot 1^2 + 20 \cdot 1 + 8 = 40$ and $k=0$. The corresponding polyomino is a $7\times 7$ square with the central $3\times 3$ square removed, containing $(2 \cdot 1 + 1)^2 = 9$ deep holes arranged in a $3\times3$ grid. This achieves $h_{40} = 9 = 40/3 + o(40)$, demonstrating the effectiveness of the “grid-of-holes” construction and the near-sharpness of the asymptotic results (Baralic et al., 11 Jan 2026).

The study of deep holes in polyominoes synthesizes discrete, topological, and geometric techniques—combining explicit lattice constructions, Pick’s theorem, and isoperimetric estimates—to characterize the extremal and enumerative properties of internal voids within polyominoes. Open challenges remain in extending these results to more diverse classes, refining enumeration, and connecting with algorithmic tiling theory (Kamenetsky et al., 2014, Baralic et al., 11 Jan 2026).