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%2F16%3A10332029" target="_blank" >RIV/00216208:11320/16:10332029 - isvavai.cz</a>

Result on the web

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.43" target="_blank" >http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.43</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.43" target="_blank" >10.4230/LIPIcs.SoCG.2016.43</a>

Result language

angličtina

Original language name

Shortest path embeddings of graphs on surfaces

Original language description

The classical theorem of Fáry 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 Fáry'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

D - Article in proceedings

CEP classification

IN - Informatics

OECD FORD branch

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Project

<a href="/en/project/GBP202%2F12%2FG061" target="_blank" >GBP202/12/G061: Center of excellence - Institute for theoretical computer science (CE-ITI)</a><br>

Continuities

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

Publication year

2016

Confidentiality

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

Article name in the collection

32nd International Symposium on Computational Geometry

ISBN

978-3-95977-009-5

ISSN

1868-8969

e-ISSN

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Number of pages

16

Pages from-to

1-16

Publisher name

Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Place of publication

Saarbrücken/Wadern, Germany

Event location

Tufts University Boston; United States

Event date

Jun 14, 2016

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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