Shortest Path Embeddings of Graphs on Surfaces

被引:4
|
作者
Hubard, Alfredo [1 ,2 ]
Kaluza, Vojtech [3 ]
de Mesmay, Arnaud [4 ]
Tancer, Martin [3 ,5 ]
机构
[1] INRIA Sophia Antipolis, F-77454 Marne La Vallee, France
[2] Univ Paris Est Marne la Vallee, F-77454 Marne La Vallee, France
[3] Charles Univ Prague, Dept Appl Math, Malostranske Namesti 25, CR-11800 Prague 1, Czech Republic
[4] CNRS, Gipsa Lab, 11 Rue Math, F-38402 St Martin Dheres, France
[5] Charles Univ Prague, Inst Theoret Comp Sci, Malostranske Namesti 25, CR-11800 Prague 1, Czech Republic
关键词
Embedded graphs; Shortest paths; Fary's theorem; Hyperbolic geometry; Graph drawing; IRREDUCIBLE TRIANGULATIONS;
D O I
10.1007/s00454-017-9898-3
中图分类号
TP301 [理论、方法];
学科分类号
081202 ;
摘要
The classical theorem of Fary states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fary's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.
引用
收藏
页码:921 / 945
页数:25
相关论文
共 50 条
  • [41] Optimal shortest path set problem in undirected graphs
    Huili Zhang
    Yinfeng Xu
    Xingang Wen
    Journal of Combinatorial Optimization, 2015, 29 : 511 - 530
  • [42] Shortest path algorithms for nearly acyclic directed graphs
    Takaoka, T
    GRAPH-THEORETIC CONCEPTS IN COMPUTER SCIENCE, 1997, 1197 : 367 - 374
  • [43] An All Pairs Shortest Path Algorithm for Dynamic Graphs
    Alshammari, Muteb
    Rezgui, Abdelmounaam
    INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER SCIENCE, 2020, 15 (01): : 347 - 365
  • [44] Faster shortest-path algorithms for planar graphs
    Henzinger, MR
    Klein, P
    Rao, S
    Subramanian, S
    JOURNAL OF COMPUTER AND SYSTEM SCIENCES, 1997, 55 (01) : 3 - 23
  • [45] Classification of reconfiguration graphs of shortest path graphs with no induced 4-cycles
    Asplund, John
    Werner, Brett
    DISCRETE MATHEMATICS, 2020, 343 (0I)
  • [46] Computing minimum distortion embeddings into a path for bipartite permutation graphs and threshold graphs
    Heggernes, Pinar
    Meister, Daniel
    Proskurowski, Andrzej
    THEORETICAL COMPUTER SCIENCE, 2011, 412 (12-14) : 1275 - 1297
  • [47] Approximate shortest path queries on weighted polyhedral surfaces
    Aleksandrov, Lyudmil
    Djidjev, Hristo N.
    Guo, Hua
    Maheshwari, Anil
    Nussbaum, Doron
    Sack, Jorg-Rudiger
    MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE 2006, PROCEEDINGS, 2006, 4162 : 98 - 109
  • [49] Number of Embeddings of Wheel Graphs on Surfaces of Small Genus
    Yang, Yan
    Liu, Yanpei
    ARS COMBINATORIA, 2011, 101 : 225 - 249
  • [50] Shortest path queries between geometric objects on surfaces
    Guo, Hua
    Maheshwari, Anil
    Nussbaum, Doron
    Sack, Joerg-Ruediger
    COMPUTATIONAL SCIENCE AND ITS APPLICATIONS - ICCSA 2007, PT 1, PROCEEDINGS, 2007, 4705 : 82 - +