Characterization of Gromov hyperbolic short graphs Articles uri icon

publication date

  • February 2014

start page

  • 197

end page

  • 212

issue

  • 2

volume

  • 30

international standard serial number (ISSN)

  • 1439-8516

electronic international standard serial number (EISSN)

  • 1439-7617

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 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.

keywords

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