Improved enumeration of simple topological graphs
| Autor: | Kynčl, Jan, Díaz Báñez, José Miguel (Coordinador), Garijo Royo, Delia (Coordinador), Márquez Pérez, Alberto (Coordinador), Urrutia Galicia, Jorge (Coordinador) |
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| Přispěvatelé: | Díaz Báñez, José Miguel, Garijo Royo, Delia, Márquez Pérez, Alberto, Urrutia Galicia, Jorge, Universidad de Sevilla. Departamento de Matemática Aplicada II |
| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Zdroj: | idUS. Depósito de Investigación de la Universidad de Sevilla instname |
| Popis: | A simple topological graph T = (V (T ), E(T )) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint 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 generalize results of Pach and Tóth and the author's previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2 O(n2log(m/n)), and at most 2O(mn1/2 log n) if m ≤ n 3/2. As a consequence we obtain a new upper bound 2 O(n3/2 log n) on the number of intersection graphs of n pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with n vertices to 2n2 ·α(n) O(1), using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize G is at most 2 m2+O(mn) and at least 2 Ω(m2) for graphs with m > (6 + ε)n. Graph Drawings and Representations, EuroGIGA Project Centre Interfacultaire Bernoulli |
| Databáze: | OpenAIRE |
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