Clusters of Cycles

A cluster of cycles (or (r; q)-polycycle) is a simple planar 2{connected nite or countable graph G of girth r and maximal vertex-degree q, which admits an (r; q)-polycyclic realization P(G) on the plane. An (r; q)-polycyclic realization is determined by the following properties: (i) all interior vertices are of degree q, (ii) all interior faces (denote their number by p r) are combinatorial r-gons, (iii) all vertices, edges and interior faces form a cell-complex.