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Suppose $G$ is connected undirected graph.

Does the calculation of the number of topologically distinct punctured surfaces that can arise from putting a ribbon structure on $G$ exist in the literature? Or in other words do we know how to list the number of punctures (or equivalently, the genera) of the possible surfaces from the knowledge of the valencies of the vertices of the graph?

Thanks.

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Think Euler characteristic. –  Jim Conant Apr 13 '11 at 22:17

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I think that Michael A. LaCroix's PhD thesis may be of interest to you. You can access it here. This is a very nicely written thesis on enumerating topological maps (Ribbon graphs are introduced in Chapter 2). The pictures in the thesis are just fantastic by the way.

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