Elementary Elliptic (r, Q)-polycycles

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