Acta Mathematica Sinica, English Series

, Volume 30, Issue 2, pp 197–212

# 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

## Supplementary material

10114_2013_2467_MOESM1_ESM.tex (58 kb)
Supplementary material, approximately 57.5 KB.

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