13,837 research outputs found
Consequences of some outerplanarity extensions
In this expository paper we revise some extensions of Kuratowski planarity criterion, providing a link between the embeddings of infinite graphs without accumulation points and the embeddings of finite graphs with some
distinguished vertices in only one face. This link is valid for any surface and for some pseudosurfaces. On the one hand, we present some key ideas that are not easily accessible. On the other hand, we state the relevance
of infinite, locally finite graphs in practice and suggest some ideas for future research
Expectation-Complete Graph Representations with Homomorphisms
We investigate novel random graph embeddings that can be computed in expected
polynomial time and that are able to distinguish all non-isomorphic graphs in
expectation. Previous graph embeddings have limited expressiveness and either
cannot distinguish all graphs or cannot be computed efficiently for every
graph. To be able to approximate arbitrary functions on graphs, we are
interested in efficient alternatives that become arbitrarily expressive with
increasing resources. Our approach is based on Lov\'asz' characterisation of
graph isomorphism through an infinite dimensional vector of homomorphism
counts. Our empirical evaluation shows competitive results on several benchmark
graph learning tasks.Comment: accepted for publication at ICML 202
A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space
A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik
(independently from each other) to metric spaces leads to a stronger version of
ultrahomogeneity of the infinite random graph , the universal Urysohn metric
space \Ur, and other related objects. We show how the result can be used to
average out uniform and coarse embeddings of \Ur (and its various
counterparts) into normed spaces. Sometimes this leads to new embeddings of the
same kind that are metric transforms and besides extend to affine
representations of various isometry groups. As an application of this
technique, we show that \Ur admits neither a uniform nor a coarse embedding
into a uniformly convex Banach space.Comment: 23 pages, LaTeX 2e with Elsevier macros, a significant revision
taking into account anonymous referee's comments, with the proof of the main
result simplified and another long proof moved to the appendi
Regular Embeddings of Canonical Double Coverings of Graphs
AbstractThis paper addresses the question of determining, for a given graphG, all regular maps havingGas their underlying graph, i.e., all embeddings ofGin closed surfaces exhibiting the highest possible symmetry. We show that ifGsatisfies certain natural conditions, then all orientable regular embeddings of its canonical double covering, isomorphic to the tensor productGâK2, can be described in terms of regular embeddings ofG. This allows us to âliftâ the classification of regular embeddings of a given graph to a similar classification for its canonical double covering and to establish various properties of the âderivedâ maps by employing those of the âbaseâ maps. We apply these results to determining all orientable regular embeddings of the tensor productsKnâK2(known as the cocktail-party graphs) and of then-dipolesDn, the graphs consisting of two vertices and n parallel edges joining them. In the first case we show, in particular, that regular embeddings ofKnâK2exist only ifnis a prime powerpl, and there are 2Ï(nâ1) orÏ(nâ1) isomorphism classes of such maps (whereÏis Euler's function) according to whetherlis even or odd. Forleven an interesting new infinite family of regular maps is discovered. In the second case, orientable regular embeddings ofDnexist for each positive integern, and their number is a power of 2 depending on the decomposition ofninto primes
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