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We introduce notions of certificates allowing to bound eccentricities in agraph. In particular , we revisit radius (minimum eccentricity) and diameter(maximum eccentricity) computation and explain the efficiency of practicalradius and diameter algorithms by the existence of small certificates forradius and diameter plus few additional properties. We show how suchcomputation is related to covering a graph with certain balls or complementaryof balls. We introduce several new algorithmic techniques related toeccentricity computation and propose algorithms for radius, diameter and alleccentricities with theoretical guarantees with respect to certain graphparameters. This is complemented by experimental results on various real-worldgraphs showing that these parameters appear to be low in practice. We alsoobtain refined results in the case where the input graph has low doublingdimension, has low hyperbolicity, or is chordal.

Revisiting Radius, Diameter, and all Eccentricity Computation in Graphs through Certificates