  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).
