Gromov hyperbolicity in lexicographic product graphs

TL;DR

This paper characterizes when the lexicographic product of two graphs is Gromov hyperbolic, showing it depends mainly on the hyperbolicity of the first graph unless it is trivial, and provides bounds on the hyperbolicity constant.

Contribution

It provides a complete characterization of hyperbolicity in lexicographic product graphs based on the properties of the factor graphs, including sharp bounds on the hyperbolicity constant.

Findings

Lexicographic product is hyperbolic iff the first factor is hyperbolic (except trivial case).

Derived bounds: elta(G_1) elta(G_1 irc G_2)  elta(G_1) + 3/2.

Characterized graphs where the upper bound on hyperbolicity is attained.

Abstract

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a {\it 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 $\delta$-\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\delta(X)=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\}.$ In this paper we characterize the lexicographic product of two graphs $G_1\circ G_2$ which are hyperbolic, in terms of $G_1$ and $G_2$: the lexicographic product graph $G_1\circ G_2$ is hyperbolic if and only if $G_1$ is hyperbolic, unless if $G_1$ is a trivial graph (the graph with a single vertex); if $G_1$ is trivial, then $G_1\circ G_2$ is hyperbolic if and only if $G_2$ is hyperbolic. In particular, we obtain the sharp inequalities $\delta(G_1)\le \delta(G_1\circ G_2) \le \delta(G_1) + 3/2$ if $G_1$ is not a trivial graph, and we characterize the graphs for which the second inequality is attained.