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The ball hypergraph of a graph $G$ is the family of balls of all possiblecenters and radii in $G$. It has Helly number at most $k$ if every subfamily of$k$-wise intersecting balls has a nonempty common intersection. A graph is$k$-Helly (or Helly, if $k=2$) if its ball hypergraph has Helly number at most$k$. We prove that a central vertex and all the medians in an $n$-vertex$m$-edge Helly graph can be computed w.h.p. in $\tilde{\cal O}(m\sqrt{n})$time. Both results extend to a broader setting where we define a non-negativecost function over the vertex-set. For any fixed $k$, we also present an$\tilde{\cal O}(m\sqrt{kn})$-time randomized algorithm for radius computationwithin $k$-Helly graphs. If we relax the definition of Helly number (for whatis sometimes called an "almost Helly-type" property in the literature), thenour approach leads to an approximation algorithm for computing the radius withan additive one-sided error of at most some constant.

Distance problems within Helly graphs and $k$-Helly graphs