## 14.1 Introduction

Given *q* ∈ and *R*⊂ , a(*R*, *q*)*polycycle P* is a nonempty 2-connected map on a closed surface *S* with faces partitioned in two nonempty sets *F*_{1} and *F*_{2}, so that:

(1) all elements of *F*_{1} (called *proper faces*) are combinatorial *i*-gons with *i* ∈ *R*;

(2) all elements of *F*_{2} (called *holes*) are pairwisely disjoint, that is, have no common vertices;

(3) all vertices have degree within {2,..., *q*} and all *interior* (i.e., not on the boundary of a hole) vertices are *q*-valent.

The map *P* can be finite or infinite and some of the faces of the set *F*_{2} can be *i*-gons with *i* ∈ *R* or have a countable number of edges. In practice almost all the maps occurring here will be plane graphs, which most often will be finite plane graphs. Note that while any finite plane graph has a unique exterior face, an infinite plane graph can have any number of exterior faces, including 0 and infinity. The exterior faces will always be holes.

An *isomorphism* between two maps, *G*_{1} and *G*_{2}, is a function φ mapping vertices, edges, and faces of *G*_{1} to those of *G*_{2} and preserving inclusion relations. Two (*R*, *q*)-polycycles, *P*_{1} and *P*_{2}, are *isomorphic* if there is an isomorphism φ of corresponding plane graphs that maps the set of holes of *P*_{1} to the set of holes of *P*_{2}. The *automorphism group* Aut(*G*) of a map *G* is the group of all ...

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