(r, q)-polycycles

A (r, q)-polycycle is a simple plane 2-connected locally finite graph with degree at most q, such that:

1. (i) all interior vertices are of degree q,

2. (ii) all interior faces are (combinatorial) r-gons.

We recall that any finite plane graph has a unique exterior face; an infinite plane graph can have any number of exterior faces, including zero and infinity. Denote by pr the number of interior faces; for example, Dodecahedron on the plane has p5 = 11.

See in Figure 4.1 some examples of connected simple plane graphs that are not (r, q)-polycycles.

We will prove later (in Theorem 4.3.2) that all vertices, edges, and interior faces of an (r, q)-polycycle form a cell-complex (see Section 1.2.1).

The skeleton of a polycycle is the edge-vertex graph defined by it, i.e. we forget the faces. By Theorem 4.3.6, except for five Platonic ones, the skeleton has a unique polycyclic realization, i.e. a polycycle for which it is the skeleton.

The parameters (r, q) are called elliptic if rq < 2(r + q), parabolic if rq = 2(r +q), and hyperbolic if rq > 2(r +q); see Remark 1.4.1. Call a polycycle outerplanar if it has no interior vertices. For parabolic or hyperbolic (r, q), the tiling {r, q} is a (r, q)-polycycle. For elliptic (r, q), the tiling {r, q} with a face deleted is an (r, q)- polycycle. Different, but all isomorphic, polycyclic realizations for those five exceptions to the unicity, come from different choices of such deleted (exterior) faces.