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).

翻译：如果一个图满足以下$α_i$-度量性质：对于任意顶点$u, w, v$和$x$，若$u$与$w$之间的最短路径和$x$与$v$之间的最短路径共享一条终端边$vw$，则$d(u,x) \ge d(u,v) + d(v,x) - i$，则该图称为$α_i$-度量图（$i \in {\cal N}$）。后者是欧几里得空间中共享终端测地线段的并集也必须是测地线这一性质的离散松弛。近期在（Dragan & Ducoffe, WG'23）中，我们首次启动了$α_i$-度量图算法应用的研究。我们此前工作中的结果与（Chepoi 等，SoCG'08）和（Chepoi 等，COCOA'18）中针对有界双曲性图所建立的结果非常相似。后者是由Gromov首次引入的、被广泛研究的度量树状性参数。在本文中，我们阐明了双曲性与$α_i$-度量性质之间的关系，证明了$α_i$-度量图是$f(i)$-双曲图，其中函数$f$关于$i$是线性的。我们利用图双曲性的多种等价定义给出了这一结果的不同证明。相比之下，我们给出了满足$1$-双曲性但并非对于任何常数$i$都为$α_i$-度量图的简单构造。最后，在$i=1$的特殊情况下，我们证明了$α_1$-度量图是$1$-双曲的，并且该界是紧的。通过这样做，我们能够回答（Dragan & Ducoffe, WG'23）中留下的一些开放问题。