We extend known results on chordal graphs and distance-hereditary graphs to much larger graph classes by using only a common metric property of these graphs. Specifically, a graph is called $\alpha_i$-metric ($i\in \mathcal{N}$) 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 $d(u,x)\geq d(u,v) + d(v,x)-i$. Roughly, gluing together any two shortest paths along a common terminal edge may not necessarily result in a shortest path but yields a ``near-shortest'' path with defect at most $i$. It is known that $\alpha_0$-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are $\alpha_i$-metric for $i=1$ and $i=2$, respectively. We show that an additive $O(i)$-approximation of the radius, of the diameter, and in fact of all vertex eccentricities of an $\alpha_i$-metric graph can be computed in total linear time. Our strongest results are obtained for $\alpha_1$-metric graphs, for which we prove that a central vertex can be computed in subquadratic time, and even better in linear time for so-called $(\alpha_1,\Delta)$-metric graphs (a superclass of chordal graphs and of plane triangulations with inner vertices of degree at least $7$). The latter answers a question raised in (Dragan, IPL, 2020). Our algorithms follow from new results on centers and metric intervals of $\alpha_i$-metric graphs. In particular, we prove that the diameter of the center is at most $3i+2$ (at most $3$, if $i=1$). The latter partly answers a question raised in (Yushmanov & Chepoi, Mathematical Problems in Cybernetics, 1991).

翻译：我们将弦图与距离遗传图的已知结果推广到更大的图类，仅利用这些图共有的度量性质。具体而言，若图对于任意顶点$u,w,v$和$x$满足如下$\alpha_i$-度量性质（$i\in \mathcal{N}$）：若一条$u$与$w$之间的最短路径和一条$x$与$v$之间的最短路径共享一条终端边$vw$，则$d(u,x)\geq d(u,v)+d(v,x)-i$。直观上，将任意两条沿公共终端边拼接的最短路径未必形成最短路径，但会生成缺陷至多为$i$的"近最短"路径。已知$\alpha_0$-度量图恰好是托勒密图，而弦图和距离遗传图分别对应$i=1$和$i=2$的$\alpha_i$-度量图。我们证明：可在总线性时间内计算$\alpha_i$-度量图的半径、直径乃至所有顶点离心率的$O(i)$加法近似值。最强的结果针对$\alpha_1$-度量图：我们证明可在次二次时间内找到中心顶点，对于所谓$(\alpha_1,\Delta)$-度量图（弦图及内点度数至少为$7$的平面三角剖分图的超类）甚至可在线性时间内完成。后者回答了(Dragan, IPL, 2020)提出的问题。我们的算法源于$\alpha_i$-度量图中心与度量区间的新结论。特别地，我们证明中心直径至多为$3i+2$（当$i=1$时，至多为$3$），该结果部分回答了(Yushmanov & Chepoi, Mathematical Problems in Cybernetics, 1991)的问题。