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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 where a is the number of equivalence classes of embeddings of G into the orientable surface S with g genera. In this paper, we compute the genus polynomial of a graph obtained from a cycle by replacing each edge by two multiple edges. [ABSTRACT FROM AUTHOR]