Elementary elliptic $(R,q)$-polycycles

Abstract

We consider the following generalization of the decomposition theorem for polycycles. A {\em $(R,q)$-polycycle} is, roughly, a plane graph, whose faces, besides some disjoint {\em holes}, are $i$-gons, $i \in R$, and whose vertices, outside of holes, are $q$-valent. Such polycycle is called {\em elliptic}, {\em parabolic} or {\em hyperbolic} if $\frac{1}{q} + \frac{1}{r} - {1/2}$ (where $r={max_{i \in R}i}$) is positive, zero or negative, respectively. An edge on the boundary of a hole in such polycycle is called {\em open} if both its end-vertices have degree less than $q$. We enumerate all elliptic {\em elementary} polycycles, i.e. those that any elliptic $(R,q)$-polycycle can be obtained from them by agglomeration along some open edges.