A Pseudo-approximation for the Genus of Hamiltonian Graphs
The genus of a graph is a very basic parameter in topological graph theory, that has been the subject of extensive study. Perhaps surprisingly, despite its importance, the problem of approximating the genus of a graph is very poorly understood. It has been shown to be NPcomplete by Thomassen [Tho89], and the best known upper bound for general graphs is an O(n)-approximation that follows by Euler’s characteristic.
We give a polynomial-time pseudo-approximation algorithm for the orientable genus of Hamiltonian graphs. More specifically, on input a graph G of orientable genus g, and a Hamiltonian path in G, our algorithm computes a drawing into a surface of either orientable, or nonorientable genus g O(1).
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