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

Abstract

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 $(\alpha _1,\varDelta )$-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).

This work was supported by a grant of the Romanian Ministry of Research, Innovation and Digitalization, CCCDI - UEFISCDI, proect number PN-III-P2-2.1-PED-2021-2142, within PNCDI III.