$\alpha_i$-Metric Graphs: Hyperbolicity

TL;DR

This paper explores the relationship between $oldsymbol{ ext{ extalpha}_i}$-metric graphs and hyperbolicity, establishing bounds and properties that connect these two graph-theoretic concepts.

Contribution

It proves that $ ext{ extalpha}_i$-metric graphs are $f(i)$-hyperbolic with a linear function $f$, and characterizes the case when $i=1$ as exactly 1-hyperbolic.

Findings

$ ext{ extalpha}_i$-metric graphs are $f(i)$-hyperbolic with linear $f$

1-$ ext{ extalpha}_1$-metric graphs are exactly 1-hyperbolic

constructed 1-hyperbolic graphs that are not $ ext{ extalpha}_i$-metric for any constant $i$

Abstract

A graph is called $\alpha_i$-metric ($i \in {\cal N}$) if it satisfies the following $\alpha_i$-metric property for every vertices $u, w, v$ and $x$: if a shortest path between $u$ and $w$ and a shortest path between $x$ and $v$ share a terminal edge $vw$, then $d(u,x) \ge d(u,v) + d(v,x) - i$. The latter is a discrete relaxation of the property that in Euclidean spaces the union of two geodesics sharing a terminal segment must be also a geodesic. Recently in (Dragan & Ducoffe, WG'23) we initiated the study of the algorithmic applications of $\alpha_i$-metric graphs. Our results in this prior work were very similar to those established in (Chepoi et al., SoCG'08) and (Chepoi et al., COCOA'18) for graphs with bounded hyperbolicity. The latter is a heavily studied metric tree-likeness parameter first introduced by Gromov. In this paper, we clarify the relationship between hyperbolicity and the $\alpha_i$-metric property, proving that $\alpha_i$-metric graphs are $f(i)$-hyperbolic for some function $f$ linear in $i$. We give different proofs of this result, using various equivalent definitions to graph hyperbolicity. By contrast, we give simple constructions of $1$-hyperbolic graphs that are not $\alpha_i$-metric for any constant $i$. Finally, in the special case of $i=1$, we prove that $\alpha_1$-metric graphs are $1$-hyperbolic, and the bound is sharp. By doing so, we can answer some questions left open in (Dragan & Ducoffe, WG'23).