## COMPUTING THE STRETCH OF AN EMBEDDED GRAPH

Autor(en): | Cabello, Sergio Chimani, Markus Hlineny, Petr |

Stichwörter: | crossings; embedded graph; homology basis; Mathematics; Mathematics, Applied; nonseparating cycle; PLANAR CROSSING NUMBERS; SURFACE; topological graph theory |

Erscheinungsdatum: | 2014 |

Herausgeber: | SIAM PUBLICATIONS |

Enthalten in: | SIAM JOURNAL ON DISCRETE MATHEMATICS |

Band: | 28 |

Ausgabe: | 3 |

Startseite: | 1391 |

Seitenende: | 1401 |

Zusammenfassung: | Let G be a graph embedded in an orientable surface Sigma, possibly with edge weights, and denote by len(gamma) the length (the number of edges or the sum of the edge weights) of a cycle. in G. The stretch of a graph embedded on a surface is the minimum of len(alpha) . len(beta) over all pairs of cycles alpha and beta that cross exactly once. We provide two algorithms to compute the stretch of an embedded graph, each based on a different principle. The first algorithm is based on surgery and computes the stretch in time O(g(4)n log n) with high probability, or in time O(g(4)n log(2) n) in the worst case, where g is the genus of the surface S and n is the number of vertices in G. The second algorithm is based on using a short homology basis and computes the stretch in time O(n(2) log n n(2)g ng(3)). |

ISSN: | 08954801 |

DOI: | 10.1137/130945636 |

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