A(*R*, *q*)*-polycycle* is a map whose faces, besides some disjoint *holes*, are *i*-gons, *i* ∈ *R*, and whose vertices have a degree between 2 and *q* with vertices outside of holes being *q*-valent. This notion arises in organic chemistry and crystallography as well as in purely mathematical contexts.

A(*R*, *q*)-polycycle is called *elementary* if it cannot be cut along an edge. Every (*R*, *q*)-polycycle can be uniquely decomposed into elementary ones. This decomposition is useful for computer enumeration and determination of classes of plane graphs and (*R*, *q*)-polycycles [6, 11–14, 17, 20]. A critical step for using the decomposition theorem is to be able to list all elementary polycycles occurring in a given problem.

A(*R*, *q*)-polycycle is called *elliptic*, *parabolic*, or *hyperbolic* if (where *r* = *max*_{i ∈}* _{R}i*) is positive, zero, or negative, respectively). Here we determine all elementary elliptic (

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