Quad-Graphs: Quadrilateral Cell Decompositions

• Quadrilateral cellular decompositions are CW-complexes where each face is a quadrilateral, linking surface topology with discrete complex analysis and integrable system models.
• The methodology employs cross‐ratio weights and discrete Cauchy–Riemann equations to establish local complex structures and convergence to classical analytic results.
• Applications extend to computer graphics and geometric modeling, with quad-layout immersions and algorithmic enumeration providing tools for discrete Riemann surface analysis.

A quadrilateral cellular decomposition, commonly known as a quad-graph, is a CW-complex decomposition of a compact 2-manifold (with or without boundary) in which every 2-cell is attached along a simple closed chain of four 1-cells, resulting in a tiling by quadrilaterals. These combinatorial structures unify perspectives from surface topology, discrete complex analysis, and integrable systems, and provide the discrete backbone for a wide range of developments—including discrete Riemann surface theory, integrable quad-equations, and the generation of quadrilateral meshes for computational and geometric applications (Cherkashyn, 2023, Bobenko et al., 2015, Bobenko et al., 2015, Düntsch et al., 31 Dec 2025).

1. Combinatorial Structure and Graph-Theoretic Foundations

A quad-graph $\Lambda$ on a closed oriented surface $\Sigma$ consists of a collection of 0-cells (vertices), 1-cells (edges), and 2-cells (faces) with each face bounded by exactly four edges. Strong regularity requires any two faces to be either disjoint or intersect in a single edge or vertex. For orientable surfaces, the Euler characteristic $\chi$ satisfies

$\chi = V - E + Q = V - Q,$

as $E = 2Q$ due to each quadrilateral contributing four edges, counted with sharing (Cherkashyn, 2023). Bipartiteness is imposed: $V(\Lambda)$ splits into black ($V(\Gamma)$) and white ($V(\Gamma^*)$) vertices, with every edge joining different colors. Associated graphs include:

• The black and white graphs $\Gamma$ and $\Gamma^*$, connecting same-color vertices across quadrilaterals.
• The dual graph $\diamond$ (vertex for each quad, edge for each shared edge).
• The medial graph $X$, whose vertices are midpoints of $\Lambda$'s edges, interlacing primal and dual combinatorics (Bobenko et al., 2015, Bobenko et al., 2015).

Quad-graphs form the cellular substrate for discrete Morse theory, with Morse–Smale flows on surfaces yielding quad-decompositions where 0-cells are foci, 2-cells correspond to saddles, and 1-cells to separatrix connections (Cherkashyn, 2023).

2. Discrete Complex Structure and Holomorphicity

The local complex structure of a quad-graph is encoded by assigning each quadrilateral $Q$ a cross-ratio weight

$$\rho_Q = -i\frac{w_+ - w_-}{b_+ - b_-}, \quad \Re(\rho_Q) > 0,$$

where $(b_-, w_-, b_+, w_+)$ are the vertices in counterclockwise order (Düntsch et al., 31 Dec 2025, Bobenko et al., 2015). Discrete holomorphic functions $f: V(\Lambda)\rightarrow\mathbb{C}$ satisfy, for every face $Q$,

$$f(w_+) - f(w_-) = i\,\rho_Q \left(f(b_+) - f(b_-)\right),$$

mirroring the Cauchy–Riemann equations in the discrete setting (Bobenko et al., 2015). Equivalently, in a chart $z_Q$,

$$\frac{f(b_+)-f(b_-)}{z_Q(b_+)-z_Q(b_-)}\ =\ \frac{f(w_+)-f(w_-)}{z_Q(w_+)-z_Q(w_-)}.$$

The strong correspondence with continuous theory is furthered by discrete versions of exterior calculus, with the discrete Hodge star, Laplacian, and Green’s identities defined using the medial graph $X$ (Bobenko et al., 2015, Bobenko et al., 2015).

3. Integrable Systems and Laplace-Type Equations

Bipartite isoradial (rhombic) quad-graphs in $\mathbb{C}$ enable the discretization of linear integrable systems via three-leg quad-equations defined on each rhombic face. These take the form

$$f(\alpha, \beta)x_{12} - g(\alpha, \beta)x_0 = i\left(h(\beta)x_1 - h(\alpha)x_2\right),$$

with parameters set by edge directions (“rapidities”) $\alpha, \beta$. Summing over all faces at a black vertex yields discrete Laplace-type equations of the form

$\sum_{j\sim i}W_{ij}(x_j-x_i) = 0,$

where the Laplacian weights $W_{ij}$ are derived from the underlying integrable structure (Bobenko et al., 2019). Integrability is characterized by multi-dimensional consistency (“3D-consistency”), reducing to a functional equation for the coefficients $f$ and $h$, whose elliptic solutions connect to the theory of discrete exponential functions and pluri-Lagrangian systems. The structure also admits a Bäcklund transformation formalism and underpins variational structures (star–triangle relations) central to discrete integrability (Bobenko et al., 2019).

4. Discrete Riemann Surfaces: Period Matrices, Involutions, and Topology

Quad-graphs form the foundation for discrete Riemann surface theory, incorporating:

• Discrete Abelian differentials and their periods, with the period matrix $\Pi$ symmetrized over black and white a- and b-periods.
• The construction of a discrete antiholomorphic involution $\tau$ (with color-preserving or color-reversing action) whose fixed-point set realizes discrete real ovals, paralleling classical real algebraic curves.
• Combinatorial derivation of a symplectic homology basis $(a_1,\ldots,a_g; b_1,\ldots,b_g)$ adapted to $\tau$, leading to the decomposition of $\Pi = \frac{1}{2}H + i T$, where $H$ encodes topological type and $T$ is real symmetric positive definite. For discrete M-curves (maximal number of ovals), this matrix is purely imaginary (Düntsch et al., 31 Dec 2025).
• Topological classification mirrors the smooth setting: Harnack’s inequality for the number of real ovals, parity constraints for dividing/non-dividing surfaces, and discrete analogues of fundamental theorems in Riemann surface theory (Düntsch et al., 31 Dec 2025, Bobenko et al., 2015).

5. Surface Classification, Labeling Schemes, and Enumeration

Every closed 2-manifold can be represented by side-pairing gluings of a $2Q+2$-gon arising from $Q$ quadrilaterals. Classification proceeds via labeling schemes: cyclic words in an alphabet $\{a_1^{\pm1}, \ldots, a_Q^{\pm1}\}$ encoding pairings and their orientations. This representation connects to the group-theoretic presentations of the fundamental group:

$$\pi_1(S) \cong \langle a_1, \ldots, a_Q \mid w = 1 \rangle,$$

where $w$ is the boundary word. Enumeration algorithms use such labelings, with brute-force backtracking and symmetry factorization yielding explicit counts for low genus (Cherkashyn, 2023). Closed formulas involve double-factorials for side pairings and dihedral group orders for symmetry reduction.

Q	Typical Surface (Orientable)	Labeling Word Example
1	Torus ($g=1$)	$a b a^{-1} b^{-1}$
3	Genus $2$ surface	$$a_1 b_1 a_1^{-1} b_1^{-1} a_2 b_2 a_2^{-1} b_2^{-1}$$
1	Sphere ($g=0$)	$a a^{-1} b b^{-1}$

6. Discrete Exterior Calculus and Analytic Structures

Analysis on quad-graphs leverages the discrete exterior differential $d$, Hodge star $\star$, codifferential $\delta$, and Laplacian $\Delta$ constructed on the medial graph $X$. This supports the construction of discrete harmonic, holomorphic, and anti-holomorphic functions, the definition of discrete Green’s functions and Cauchy kernels, and proves discrete analogues of classical results such as Green’s identities, Morera’s theorem, and the Riemann–Roch theorem (Bobenko et al., 2015, Bobenko et al., 2015). For planar parallelogram-graphs with uniform geometry, discrete solutions converge to classical analytic objects under mesh refinement.

7. Applications and Quad-Layout Immersions

Quadrilateral layouts, and their immersion representations, are instrumental in computer graphics and geometric modeling. "Quad-layout immersions" give a mathematically equivalent formulation to surface quadrilateral meshes by isometrically immersing a simply connected cut-open surface into $\mathbb{R}^2$ via global orthogonal coordinate one-forms. These coordinate maps relate to the existence of flat cone metrics with $\pi/2$ holonomy and establish equivalence with quartic differential structures on surfaces (Shepherd et al., 2020). Discrete realizations (integer-grid maps) are computed via linear-algebraic solvers on triangular meshes, aligning quad-mesh extraction with variational and combinatorial methods.

The quad-graph paradigm thus synthesizes topology, discrete complex analysis, integrable system theory, and algorithmic enumeration, presenting a unified framework for both theoretical advances and computational applications in discrete differential geometry and the analysis of Riemann surfaces (Cherkashyn, 2023, Bobenko et al., 2015, Bobenko et al., 2015, Düntsch et al., 31 Dec 2025, Bobenko et al., 2019, Shepherd et al., 2020).