$$\alpha _i$$-Metric Graphs: Radius, Diameter and all Eccentricities
Résumé
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).
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Licence : CC BY - Paternité
Licence : CC BY - Paternité