Hyperbolicity in median graphs |
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Authors: | JOSÉ M SIGARRETA |
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Institution: | 1. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650, Acalpulco Gro., Mexico
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Abstract: | If X is a geodesic metric space and x 1,x 2,x 3?∈?X, a geodesic triangle T?=?{x 1,x 2,x 3} is the union of the three geodesics x 1 x 2], x 2 x 3] and x 3 x 1] in X. The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X. If X is hyperbolic, we denote by δ(X) the sharp hyperbolicity constant of X, i.e., ${\delta}(X)=\inf\{{\delta}\ge 0: \, X \, \text{is $\delta$-hyperbolic}\}. $ In this paper we study the hyperbolicity of median graphs and we also obtain some results about general hyperbolic graphs. In particular, we prove that a median graph is hyperbolic if and only if its bigons are thin. |
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