${\alpha }_i$-Metric Graphs: Radius, Diameter and all Eccentricities

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 $d(u,x)\ge d(u,v)+d(v,x)-i$. 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 (Inf Probl Lett 154:105873, 2020), 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 and Chepoi (Math Probl Cybernet 3:217–232, 1991).