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).

翻译：如果一个图满足以下对任意顶点$u, w, v$和$x$的$\alpha_i$-度量性质：若$u$与$w$间的一条最短路径和$x$与$v$间的一条最短路径共享一条末端边$vw$，则$d(u,x) \ge d(u,v) + d(v,x) - i$，则称该图是$\alpha_i$-度量的（$i \in {\cal N}$）。后者是欧氏空间中两条共享末端段的测地线的并集必然也是测地线这一性质的离散松弛。近期在(Dragan & Ducoffe, WG'23)中，我们首次开展了$\alpha_i$-度量图算法应用的研究。我们在该先前工作中的结果与(Chepoi等人, SoCG'08)和(Chepoi等人, COCOA'18)中针对有界双曲性图建立的结果高度相似。后者是由Gromov首次引入的、被广泛研究的度量树状参数。本文澄清了双曲性与$\alpha_i$-度量性质之间的关系，证明$\alpha_i$-度量图是$f(i)$-双曲的，其中函数$f$关于$i$线性。我们采用图双曲性的多种等价定义，给出了该结果的不同证明。相比之下，我们构造了简单的例子，表明存在$1$-双曲图对于任意常数$i$都不是$\alpha_i$-度量的。最后，在$i=1$的特殊情形下，我们证明$\alpha_1$-度量图是$1$-双曲的，且该界是紧的。借此，我们回答了(Dragan & Ducoffe, WG'23)中遗留的一些开放问题。