\(\alpha _i\)-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 (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).