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

Abstract

AbstractWe 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 $$\alpha _i$$ α i -metric ($$i\in {\mathcal {N}}$$ i ∈ N ) if it satisfies the following $$\alpha _i$$ α 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 $$d(u,x)\ge d(u,v) + d(v,x)-i$$ 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 $$\alpha _0$$ α 0 -metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are $$\alpha _i$$ α i -metric for $$i=1$$ i = 1 and $$i=2$$ 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 $$\alpha _i$$ α i -metric graph can be computed in total linear time. Our strongest results are obtained for $$\alpha _1$$ α 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 $$(\alpha _1,\varDelta )$$ ( α 1 , Δ ) -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 $$\alpha _i$$ α i -metric graphs. In particular, we prove that the diameter of the center is at most $$3i+2$$ 3 i + 2 (at most 3, if $$i=1$$ i = 1 ). The latter partly answers a question raised in Yushmanov and Chepoi (Math Probl Cybernet 3:217–232, 1991).