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∈R
, and whose vertices, outside of holes, are
q
-valent. Such polycycle is called {\em elliptic}, {\em parabolic} or {\em hyperbolic} if
1
q
+
1
r
−1/2
(where
r=ma
x
i∈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.