αi-Metric Graphs: Radius, Diameter and all Eccentricities

We extend known results on chordal graphs and distancehereditary 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 is an element of 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) >= 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).