Distortion of the Hyperbolicity Constant of a Graph

Abstract

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this paper is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G\setminus e$ obtained from the graph $G$ by deleting an arbitrary edge $e$ from it. These inequalities allow to obtain the other main result of this paper, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity.

On the Hyperbolicity Constant of Line Graphs

“…The study of mathematical properties of Gromov hyperbolic spaces and its applications is a topic of recent and increasing interest in graph theory; see, for instance [3,4,5,9,10,11,20,28,29,30,31,32,33,36,37,39,40,41,42,46,47,48,50,52].…”

Gromov hyperbolicity in lexicographic product graphs

“…In [7] the authors obtain quantitative information about the distortion of the hyperbolicity constant of the graph G \ e obtained from the graph G by deleting an arbitrary edge e from it. The following theorem is a weak version of their main result.…”

“…The study of Gromov hyperbolic graphs is a subject of increasing interest in graph theory; see, e.g., [2,3,4,5,6,7,9,11,15,20,21,22,24,26,27,29,30,31,32,37,39] and the references therein.…”

“…Therefore, it is interesting to study the invariance of the hyperbolicity of graphs under appropriate transformations. The invariance of the hyperbolicity under some natural transformations on graphs have been studied in previous papers, for instance, removing edges of a graph is studied in [4,7], the line graph of a graph in [9,11], the dual of a planar graph in [8,29] and the complement of a graph in [5].…”

“…. , v8 } contains neither [v 1 , v 4 ], [v 1 , v 5 ], [v 2 , v 4 ] nor [v 2 , v 5 ]; besides, since d G (p, [yz] ∪ [zx]) > 1 we have that {v 1 , . .…”