Enumeration of simple complete topological graphs |
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Institution: | 1. Department of Computer Science, University of Texas at Dallas, 800 West Campbell Road Richardson, TX 75080, USA;2. Department of Applied Mathematics II, University of Seville, Camino de los Descubrimientos s/n, Seville 41092, Spain |
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Abstract: | A simple topological graph is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is . We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and crossings is at least , which improves the estimate of Harborth an Mengersen. |
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