Acta Mathematica Sinica, English Series

, Volume 30, Issue 2, pp 197–212 | Cite as

Characterization of Gromov hyperbolic short graphs

Article

Abstract

To decide when a graph is Gromov hyperbolic is, in general, a very hard problem. In this paper, we solve this problem for the set of short graphs (in an informal way, a graph G is r-short if the shortcuts in the cycles of G have length less than r): an r-short graph G is hyperbolic if and only if S9r(G) is finite, where SR(G):= sup{L(C): C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if dC(x, y) ≤ dG(x, y) + R for every x, yC.

Keywords

Short graph Gromov hyperbolicity Gromov hyperbolic graph infinite graphs geodesics 

MR(2010) Subject Classification

05C10 05C12 05C75 05A20 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Carlos III de MadridLeganés, MadridSpain

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