Genus polynomials of cycles with double edges

Abstract

Two cellular embeddings i: G → S and j: G → S of a connected graph G into a closed orientable surface S are equivalent if there is an orientation-preserving surface homeomorphism h: S → S such that hi = j. The genus polynomial of a graph G is defined by $$ g\left[ G \right](x) = \sum\limits_{g = 0}^\infty {a_g x^g ,} $$ where ag is the number of equivalence classes of embeddings of G into the orientable surface Sg with g genera.