Enumerating planar stuffed maps as hypertrees of mobiles

A planar stuffed map is an embedding of a graph into the 2-sphere $S^{2}$, considered up to orientation-preserving homeomorphisms, such that the complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to $S^{2}$ with multiple boundaries. This is a generalization of planar maps whose complement of the graph is a collection of disjoint topologically connected components that are each homeomorphic to a disc. The main goal of this work is to construct a bijection between bipartite planar stuffed maps and collections of integer-labelled trees connected by hyperedges such that they form a hypertree, called hypermobiles. This bijection directly generalizes the Bouttier-Di Franceso-Guitter bijection between bipartite planar maps and mobiles. Additionally, we show that the generating functions of these trees of mobiles satisfy both an algebraic equation, generalizing the case of ordinary planar maps, and a new functional equation. As an example, we explicitly enumerate a class of stuffed quadrangulations.

💡 Research Summary

The paper introduces a bijective correspondence between bipartite planar stuffed maps and a new combinatorial object called hyper‑mobiles, which are collections of integer‑labelled plane trees (mobiles) linked together by hyperedges forming a hypertree. A planar stuffed map is an embedding of a graph on the sphere whose complementary regions are not simple disks but elementary planar 2‑cells that may have multiple boundaries. Each such 2‑cell can be thought of as a “branch” connecting several connected components of the map; the collection of components and branches naturally defines a hypergraph, which is in fact a hypertree because any cycle would violate planarity.

The authors first recall the classic Bouttier‑Di Francesco‑Guitter (BDFG) bijection between bipartite planar maps and mobiles, and then generalize this construction. A hyper‑mobile consists of several mobiles, each having a distinguished white vertex of minimal label (the gate vertex). Hyperedges connect gate vertices of distinct mobiles; the label of a gate vertex is exactly one larger than the label of the gate vertex it is linked to within the same hyperedge. This rule preserves the distance‑label structure that underlies the original BDFG bijection while allowing the additional connectivity introduced by multi‑boundary cells.

Two explicit transformations are defined. The forward map Φ takes a rooted, pointed stuffed map and produces a hyper‑mobile by (1) recolouring all original vertices white, (2) inserting a hyperedge for each multi‑boundary 2‑cell between the “spurious points” placed on each boundary, (3) adding a black vertex inside each face (corresponding to a boundary) and connecting it to the white vertex of maximal distance on its left, and (4) erasing the original edges and shifting labels so that the root’s source has label zero. The inverse map Ψ reconstructs the stuffed map from a hyper‑mobile by adding an extra white vertex of label one less than the minimum label, reconnecting corners according to the label‑decrement rule, removing the original black vertices, and finally replacing each hyperedge with a 2‑cell whose boundaries match the incident mobiles. The authors prove that Φ and Ψ are mutual inverses, establishing a bijection.

On the enumerative side, the generating function W(t) for rooted mobiles satisfies the same algebraic equation as in the classical case, (W = t,\Phi(W)), where (\Phi) encodes the possible degrees of black vertices. For stuffed maps, an additional functional equation (F(z) = G(F(z),z)) appears, reflecting the contribution of hyperedges and multi‑boundary cells. The variables in these equations keep track of the lengths of boundaries and the arities of hyperedges, allowing one to extract coefficients for maps with prescribed specifications.

To illustrate the method, the authors focus on “stuffed quadrangulations,” a class where every 2‑cell is either a quadrangle (boundary length 4) or a two‑boundary cell with each boundary of length 2. In this setting the algebraic and functional equations simplify dramatically, yielding explicit closed‑form formulas for the number of such maps with a given number of vertices. The resulting numbers exhibit a Catalan‑like growth, confirming that the hyper‑mobile framework correctly captures the combinatorial complexity of stuffed maps.

The paper concludes by emphasizing that hyper‑mobiles provide a natural tree‑like encoding for stuffed maps, opening the way to probabilistic studies of random stuffed maps and their scaling limits. Since stuffed maps arise in multi‑trace matrix models, tensor models, and the description of wormholes in two‑dimensional quantum gravity, the bijection offers a combinatorial bridge between these physical theories and rigorous random geometry. Future work is suggested on extending the bijection to higher genus, incorporating additional decorations (e.g., matter fields), and exploring continuum limits via the derived generating functions.