Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L'angi and Szab'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.

💡 Research Summary

The paper investigates the generation of spherical multiquadrangulations—graphs embedded on the sphere whose faces are bounded by closed walks of length four—through a family of graph operations known as vertex splittings. A vertex splitting replaces a vertex v of degree D by two new vertices v₁ and v₂, redistributes the incident edges, and inserts a new edge (v₁,v₂) so that all faces remain quadrangular. The authors introduce the notion of the “degree” of a splitting (the degree D of the split vertex) and consider restricted families S_{i,j} = {splittings with i ≤ D ≤ j}. While it is known that the unrestricted family S_{2,3} generates every simple quadrangulation from the trivial seed P₂ (a path of length two), the present work focuses on the more constrained families S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, and S_{3,3}.

A central result is that the splittings in S_{1,2} are exactly the monotone splittings: after applying such a splitting the resulting graph contains the original as a subgraph. This monotonicity implies that any quadrangulation can be reduced by repeatedly applying inverse S_{1,2} operations until no further reduction is possible. The irreducible graphs obtained in this way are called ancestors. The authors prove that every quadrangulation possesses a unique ancestor with respect to S_{1,2}, and that there exist non‑trivial ancestors beyond the seed P₂. Consequently, the whole class of spherical multiquadrangulations can be partitioned into families rooted at distinct ancestors, each family being generated uniquely by a sequence of monotone splittings.

The combinatorial framework is then linked to the geometry of convex bodies. For a smooth, strictly convex body, the set of static equilibrium points on its surface can be classified into stable (s) and unstable (u) points. By drawing the gradient flow lines between them one obtains a 2‑coloured quadrangulation of the sphere, where the two colour classes correspond to the stable and unstable equilibria. The pair (s,u) uniquely determines the primary equilibrium class introduced by Várkonyi and Domokos. The paper shows that the restricted families S_{1,1} (splittings of degree‑1 vertices) and S_{2,2} (splittings of degree‑2 vertices) are sufficient to generate all possible (s,u) pairs from a finite set of ancestors. These ancestors turn out to be closely related to the minimal polyhedra (tetrahedron, cube, octahedron, etc.) that appear in the geometric literature as extremal shapes for equilibrium counts.

When the full topology of the quadrangulation is taken into account, one obtains the secondary (or refined) equilibrium classes. These classes distinguish not only the numbers of stable and unstable points but also the exact adjacency pattern among them. Recent work by Domokos, Lángi, and Szabó demonstrated that unrestricted vertex splittings can generate every secondary class. In contrast, the present study proves that monotone splittings S_{1,2} can only produce a limited subset of secondary classes, all of which share a common ancestor. The additional ancestors identified by monotone splittings correspond geometrically to minimal polyhedra, indicating that such polyhedra play a pivotal role in the limited expansion of secondary classes under monotone operations.

To substantiate the theoretical findings, the authors implemented exhaustive enumeration algorithms for multiquadrangulations and secondary equilibrium classes. Computational results confirm that the number of distinct secondary classes grows rapidly with the number of ancestors, yet remains bounded when only S_{1,2} is allowed. The data also illustrate that the combined families S_{1,1} and S_{2,2} cover the entire space of primary classes, whereas generating the full spectrum of secondary classes inevitably requires the unrestricted splitting operations.

In summary, the paper provides a detailed combinatorial classification of spherical multiquadrangulations via restricted vertex splittings, establishes the existence of unique ancestors under monotone splittings, and translates these graph‑theoretic insights into the language of mechanical equilibria of convex bodies. It clarifies which equilibrium classes can be reached by limited operations, highlights the special status of minimal polyhedra as additional ancestors, and offers extensive computational evidence supporting the theoretical claims. The work opens avenues for further exploration of higher‑dimensional analogues, alternative face sizes, and experimental validation through physical models of convex bodies.