  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).
