The topological model of generalized maps
Jerboa uses the topological model of generalized maps (or G-maps) [10] because its mathematical definition can be easily encoded within a formal framework. In G-maps, the topological structure is handled by both the graph structure with the arc labels, while the embedding is defined by the node labels.
G-map decomposition
G-maps intuitively come from object decomposition into topological cells. The 2D object of Fig. 1 is first decomposed into faces in Fig. 2 connected along their common edge with a 2-relation and provided with 2-loops on border edges. Similarly, faces are split into edges connected with the 1-relation of Fig. 3. At last, edges are split into vertices by the 0-relation to obtain the 2-G-map of Fig. 4. Nodes obtained at the end of the process are the G-map darts and the different i-relations are labeled arcs: for an n-dimensional G-map, i belongs to [0, n]. At last, the multiple embedding informations such as vertex points and face colors are encoded by multiple node labels [4] in Fig. 5.
Orbits
Topological cells of G-maps are defined by subgraphs called orbits and built from an originating node v and a set of labels o. The orbit ⟨o⟩(v) (of type ⟨o⟩ adjacent to v) is the subgraph which contains v, the nodes reachable from node v using arcs labeled on o, and the arcs themselves. For example, the vertex adjacent to e is the orbit ⟨1 2⟩(e) of Fig. 6, the ajacent edge is the orbit ⟨0 2⟩(e) of Fig. 7, and the ajacent face is the orbit ⟨0 1⟩(e) of Fig. 8. Note that by construction, embedding informations (points or colors) are shared by all nodes belonging to orbits associated to the considered embedding. For example, all nodes of the vertex of Fig. 6 are labeled with the point B, and all nodes of the face of Fig. 8 are labeled with the blue color. Note that orbits are not limited to cells and any subset [0,n] can define a valid orbit of a n-G-map. For instance, the orbit ⟨0⟩(e) of Fig. 9 defines the half-edge ajacent to e.
