α_i-Metric Graphs: Radius, Diameter and all Eccentricities
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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 α_i-metric (i∈𝒩) if it satisfies the following 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)≥ 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 α_0-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are α_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 α_i-metric graph can be computed in total linear time. Our strongest results are obtained for 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 (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, IPL, 2020). Our algorithms follow from new results on centers and metric intervals of α_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).
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