# Shortest Path Embeddings of Graphs on Surfaces

@article{Hubard2016ShortestPE,
title={Shortest Path Embeddings of Graphs on Surfaces},
author={Alfredo Hubard and Vojtech Kaluza and Arnaud de Mesmay and Martin Tancer},
journal={Discrete \& Computational Geometry},
year={2016},
volume={58},
pages={921-945}
}
• Published 22 February 2016
• Mathematics
• Discrete & Computational Geometry
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…
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## References

SHOWING 1-10 OF 61 REFERENCES
Two maps on one surface
• Mathematics
J. Graph Theory
• 2001
If one toroidal map is replaced by its mirror image, then it is shown that minimum number of crossings can decrease, but not by too much.
Discrete Systolic Inequalities and Decompositions of Triangulated Surfaces
• Mathematics
SoCG
• 2014
This work builds upon Riemannian systolic inequalities, which bound the minimum length of non-trivial closed curves in terms of the genus and the area of the surface, and proves the existence of pants decompositions of length O(g3/2n1/2) for any triangulated combinatorial surface of genus g with n triangles.
Graphs on Surfaces
• Mathematics
Johns Hopkins series in the mathematical sciences
• 2001
This chapter discusses Embeddings Combinatorially, Contractibility, of Cycles, and the Genus Problem, which focuses on planar graphs and the Jordan Curve Theorem, and colorings of Graphs on Surfaces, which are 5-choosable.
Optimal pants decompositions and shortest homotopic cycles on an orientable surface
• Mathematics
JACM
• 2007
This paper considers the problem of finding a shortest cycle (freely) homotopic to a given simple cycle on a compact, orientable surface and describes two algorithms for extending a cycle to a pants decomposition.
Tightening non-simple paths and cycles on surfaces
• Mathematics, Computer Science
SODA '06
• 2006
It is proved that the recent algorithms of Colin de Verdière and Lazarus for shortening embedded graphs and sets of cycles have running times polynomial in the complexity of the surface and the input curves, regardless of thesurface geometry.
On Embedding a Graph in the Grid with the Minimum Number of Bends
• R. Tamassia
• Computer Science, Mathematics
SIAM J. Comput.
• 1987
An algorithm is presented that computes a region preserving grid embedding with the minimum number of bends in edges with use of network flow techniques, and runs in time $O(n^2 \log n)$, where n is the number of vertices of the graph.
Crossing numbers of graph embedding pairs on closed surfaces
We shall prove that any two graphs G1 and G2 can be embedded together on a closed surface of genus g with at most 4g · β(G1) · β(G2) crossing points on their edges if they are embeddable on the
Pants Decompositions of Random Surfaces
• Mathematics
• 2010
Our goal is to show, in two different contexts, that “random” surfaces have large pants decompositions. First we show that there are hyperbolic surfaces of genus g for which any pants decomposition
Embeddability in the 3-Sphere Is Decidable
• Mathematics, Computer Science
J. ACM
• 2018
We show that the following algorithmic problem is decidable: given a 2-dimensional simplicial complex, can it be embedded (topologically, or equivalently, piecewise linearly) in R3? By a known
Untangling two systems of noncrossing curves
• Mathematics
Graph Drawing
• 2013
It is proved that if $\mathcal{M}$ is planar, i.e., a sphere with hi¾?0 boundary components "holes", then Omn crossings can be achieved independently of h, which is asymptotically tight, as an easy lower bound shows.