Minimal Vertex Model

• Minimal vertex models are mathematical constructs that minimize the number or complexity of vertices while preserving key geometric, topological, or energetic properties.
• They are applied in discrete isoperimetry and triangulations, providing explicit formulas for minimal boundaries and unique constructions in manifolds like RP⁴.
• Their use extends to polyhedral embeddings and biophysical tissue mechanics, exemplified by the 8-vertex paper torus and energy-minimizing models in cell sheets.

A minimal vertex model is a construct in mathematics, combinatorics, and mathematical physics where the cardinality of a set of vertices, or the variables associated with vertices, is minimized under certain structural, topological, or functional constraints. These models arise across discrete geometry, statistical mechanics, combinatorial topology, and biophysical modeling. Distinct contexts share the central theme: minimizing either the count or the combinatorial complexity of vertex sets while preserving specified geometric, topological, or energetic properties.

1. Minimal Vertex Models in Discrete Isoperimetry

In discrete geometry, the minimal vertex model refers to the characterization of subsets of the planar integer lattice $X = \mathbb{Z}^2_{\ell_1}$ that minimize the size of the vertex boundary for a given cardinality. Formally, for a finite subset $S \subset X^0$, the vertex boundary is

$$\partial S = \{ v \in X^0 \setminus S \mid \text{there is an edge from } v \text{ into } S \}.$$

A set $S$ is minimal of size $n$ if $|\partial S|$ is minimized among all such subsets of size $n$ (Gupta et al., 2020).

A geometric classification emerges: saturated minimal sets are "boxes" bounded by lines of slopes $1$ and $-1$, up to congruence, taking the form

$$B(\alpha,\beta) = \{ (x,y) \in \mathbb{Z}^2 \mid 0 \leq y - x \leq \alpha,\ 0 \leq y + x \leq \beta \}$$

or, for certain parity, "half-offset" boxes. Minimal sets are always saturated, and non-box minimal sets are obtained by peeling vertices along quadrant cones from the bounding box. The boundary and area of each box have explicit formulas: $$|\partial B(\alpha, \beta)| = \alpha + \beta + 4,\quad |B(\alpha,\beta)| = \left\lfloor \frac{\alpha\beta + \alpha + \beta + 2}{2} \right\rfloor.$$

The concept of excess quantifies possible deletions while preserving minimal boundary: $$\mathrm{Exc}(B(\alpha,\beta)) = \left\lfloor \frac{\lfloor (\alpha + \beta)/2\rfloor - ((\beta - \alpha)/2)^2}{2} \right\rfloor.$$ Minimal sets maximize occupancy in their bounding box except for the removal of excess vertices forming discrete cones (Gupta et al., 2020).

2. Minimal Vertex Triangulations of Manifolds

The minimal vertex count required to triangulate a given combinatorial manifold, particularly for high-dimensional or projective spaces, is a fundamental problem in combinatorial topology. For example, for the 4-dimensional real projective space $\mathbb{R}P^4$, the Arnoux–Marin bound asserts that any simplicial triangulation requires at least $f_0 = \binom{6}{2} + 1 = 16$ vertices. Explicit constructions realize this bound, yielding a vertex-minimal triangulation $\mathrm{RP}^4_{16}$, which is unique up to isomorphism (Balagopalan, 2014).

Multiple construction schemes exist:

• Double covers of an antipodal $S^4$ manifold with barycentric subdivisions and quotienting by an involution yield 16-vertex triangulations.
• Triangulations via cross-polytopes and their dual hypercubes, with a process of bistellar flips and identification, achieve the same minimality.
• Mixed fillings of a suspended cube and octahedral prism, followed by antipodal identification, provide yet another construction (Balagopalan, 2014).

The triangulation's $f$-vector is $(16,120,380,450,150)$, and the automorphism group is isomorphic to $S_6$. The structure interfaces with the theory of block designs, as seen in the connection to the Witt design $W_{22}$.

3. Vertex-Minimal Models in Polyhedral Embeddings: The 8-Vertex Paper Torus

In the setting of isometric polyhedral embeddings, the minimal vertex model addresses the fewest-vertex realization of a given metric structure. The archetypal result is the embedding of flat tori into $\mathbb{R}^3$ as piecewise-linear isometric surfaces ("paper tori") (Doyle et al., 14 Oct 2025).

Key results:

• Any PL isometric embedding of a flat torus into $\mathbb{R}^3$ requires at least eight vertices due to curvature and vertex-degree constraints (via Gauss–Bonnet-type arguments).
• There exists an explicit uniform triangulation with 8 vertices, each of degree 6, admitting embeddings whose combinatorial structure is parameterized by a modulus $z$ in a bi-cusped fundamental domain.
• The near universality theorem states that every flat torus without reflection symmetries can be embedded as an 8-vertex paper torus. For arbitrary $\epsilon > 0$, any flat torus can be approximated in the Hausdorff metric by a suitable 8-vertex paper torus.
• The moduli space of these embeddings is 6-dimensional, with an open path-connected subset mapping surjectively onto the moduli space of tori without reflection. The closure encompasses all flat tori (possibly as immersed, rather than embedded, surfaces) (Doyle et al., 14 Oct 2025).

4. Minimal Vertex Models in Biophysical Tissue Mechanics

Vertex models, in biophysics, formulate the energy of a confluent two-dimensional cell sheet as a function of cell perimeters and areas. The minimal vertex model in this context seeks the least-complex set of rules and parameters consistent with experimental data, often by minimizing the number of free parameters or controlling polydispersity.

For Drosophila amnioserosa during dorsal closure, a minimal vertex model is constructed as follows (Tah et al., 2023):

• The tissue is modeled as a network of $N$ cells, each cell $i$ characterized by area $a_i$ and perimeter $p_i$, with energy

$$E = \sum_{i=1}^N \left[ \frac{1}{2}k_a (a_i - a_0)^2 + \frac{1}{2}k_p (p_i - p_{0,i})^2 \right].$$

• The model incorporates initial perimeter polydispersity (random $p_{0,i}$) drawn from a distribution matching experimental shape indices.
• The key parameter is a time-dependent shrinkage of each $p_{0,i}(t)$, proportional to the global decrease in tissue area ($\Delta A$).
• This framework prevents fluidization (absence of T1 transitions) throughout morphogenesis, accurately reproducing empirical metrics of cell shape, orientation, and junction tension (Tah et al., 2023).

5. Special Classes, Structural Graphs, and Generalizations

Minimal vertex models also categorize special subclasses and encode their relationships through structural graphs:

• Uniquely minimal sets: Sizes for which there exists a unique (up to congruence) minimal set; in lattice models, these are the Wang–Wang boxes $B(2m,2m)$ or $B(m,m+1)$.
• Efficient sets: For a fixed boundary size, sets maximizing area, coinciding with boxes $B(m,m)$, $B(m,m+1)$, or $B(2m,2m+2)$.
• Mortal, dead, and immortal sets: Classification by the potential for growth while preserving minimality; immortality corresponds to efficient boxes.

The graph of minimal sets, $\mathcal{G}$, has as vertices the congruence classes of minimal sets, with edges for single-vertex transitions. This graph possesses one infinite component (containing the Wang–Wang sequence), infinitely many isolated vertices (corresponding to zero-excess boxes), and finite components of arbitrarily large height (Gupta et al., 2020).

The structural framework extends to higher-dimensional lattices, edge-isoperimetry, and network theory, demonstrating the general utility of minimal vertex models in combinatorics and beyond.

6. Key Algebraic and Combinatorial Ingredients

Explicit formulas underpin the structure and universality of minimal vertex models:

• For lattice minimality, closed forms exist for boundary size, area, and excess as parities of box parameters.
• In the 8-vertex paper torus, cone-angles as functions of selected height coordinates, with the determinant of the Jacobian,

$$\det(dF(x,y)) = -\frac{64\sqrt{2}x^{3/2}\left(2x^2\gamma_0 + x^2\gamma_3 + y^2\gamma_3\right)}{\gamma_3^4 (2x\gamma_0 + (2x+1)(x^2+y^2))},$$

control local diffeomorphism and moduli regularity. Embedding robustness is guaranteed by ensuring all tetrahedral volumes have strict sign patterns throughout deformation (Doyle et al., 14 Oct 2025).

The unifying theme is the reduction of combinatorial or geometric complexity (vertex count or variability) subject to stringent topological or physical constraints, with explicit realization in multiple domains and rigorous mathematical control over all admissible configurations.