Shortest Path Embeddings of Graphs on Surfaces

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F17%3A10366180" target="_blank" >RIV/00216208:11320/17:10366180 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.1007/s00454-017-9898-3" target="_blank" >http://dx.doi.org/10.1007/s00454-017-9898-3</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00454-017-9898-3" target="_blank" >10.1007/s00454-017-9898-3</a>

Result language

angličtina

Original language name

Shortest Path Embeddings of Graphs on Surfaces

Original language description

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&apos;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.

Czech name

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Czech description

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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

Result was created during the realization of more than one project. More information in the Projects tab.

Continuities

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year

2017

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Discrete and Computational Geometry

ISSN

0179-5376

e-ISSN

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Volume of the periodical

58

Issue of the periodical within the volume

4

Country of publishing house

US - UNITED STATES

Number of pages

25

Pages from-to

921-945

UT code for WoS article

000413958900008

EID of the result in the Scopus database

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