$α_i$-Metric Graphs: Hyperbolicity

A graph is called $α_i$-metric ($i \in {\cal N}$) if it satisfies the following $α_i$-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 $α_i$-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 $α_i$-metric property, proving that $α_i$-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 $α_i$-metric for any constant $i$. Finally, in the special case of $i=1$, we prove that $α_1$-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).