Characterization of Gromov hyperbolic short graphs Articles uri icon

publication date

  • February 2014

start page

  • 197

end page

  • 212


  • 2


  • 30

International Standard Serial Number (ISSN)

  • 1439-8516

Electronic International Standard Serial Number (EISSN)

  • 1439-7617


  • 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 S (9r) (G) is finite, where S (R) (G):= sup{L(C): C is an R-isometric cycle in G} and we say that a cycle C is R-isometric if d (C) (x, y) a parts per thousand currency sign d (G) (x, y) + R for every x, y a C.


  • short graph; gromov hyperbolicity; gromov hyperbolic graph; infinite graphs; geodesics; quasihyperbolic metrics; denjoy domains; constant; surfaces; spaces