1,368 research outputs found
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry 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 F\'{a}ry'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.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Complex Networks on Hyperbolic Surfaces
We explore a novel method to generate and characterize complex networks by
means of their embedding on hyperbolic surfaces. Evolution through local
elementary moves allows the exploration of the ensemble of networks which share
common embeddings and consequently share similar hierarchical properties. This
method provides a new perspective to classify network-complexity both on local
and global scale. We demonstrate by means of several examples that there is a
strong relation between the network properties and the embedding surface.Comment: 8 Pages 3 Figure
Embedding of metric graphs on hyperbolic surfaces
An embedding of a metric graph on a closed hyperbolic surface is
\emph{essential}, if each complementary region has a negative Euler
characteristic. We show, by construction, that given any metric graph, its
metric can be rescaled so that it admits an essential and isometric embedding
on a closed hyperbolic surface. The essential genus of is the
lowest genus of a surface on which such an embedding is possible. In the next
result, we establish a formula to compute . Furthermore, we show that
for every integer , admits such an embedding (possibly
after a rescaling of ) on a surface of genus .
Next, we study minimal embeddings where each complementary region has Euler
characteristic . The maximum essential genus of is
the largest genus of a surface on which the graph is minimally embedded.
Finally, we describe a method explicitly for an essential embedding of , where and are realized.Comment: Revised version, 11 pages, 3 figure
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