A story of diameter, radius and (almost) Helly property

We present new algorithmic results for the class of Helly graphs, i.e., for the discrete analogues of hyperconvex metric spaces. Specifically, an undirected unweighted graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. It is known that every graph isometrically embeds into a Helly graph, that makes of the latter an important class of graphs in Metric Graph Theory. We study diameter and radius computations within the Helly graphs, and related graph classes. This is in part motivated by a conjecture on the finegrained complexity of these two distance problems within the graph classes of bounded fractional Helly number — that contain as particular cases the proper minor-closed graph classes and thebounded clique-width graphs. Note that under plausible complexity assumptions, neither thediameter nor the radius can be computed in truly subquadratic time on general graphs.