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, 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).
This work was supported by a grant of the Romanian Ministry of Research, Innovation and Digitalization, CCCDI - UEFISCDI, proect number PN-III-P2-2.1-PED-2021-2142, within PNCDI III.
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Notes
- 1.
It is conjectured in [39] that \(diam(C(G))\le i+2\) for every \(\alpha _i\)-metric graph G.
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Dragan, F.F., Ducoffe, G. (2023). \(\alpha _i\)-Metric Graphs: Radius, Diameter and all Eccentricities. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_20
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