Polycycles
Michel Marie Deza, Mathieu Dutour Sikirić · Cambridge University Press eBooks · 2008
( r, q )-polycycles A ( r, q ) -polycycle is a simple plane 2-connected locally finite graph with degree at most q , such that: (i) all interior vertices are of degree q , (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 p r the number of interior faces; for example, Dodecahedron on the plane has p 5 = 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 ); 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.