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The δ-hyperbolicity of a graph is defined by a simple 4-point condition: for any four vertices u, v, w, and x, the two larger of the distance sums d(u, v) + d(w, x), d(u, w) + d(v, x), and d(u, x) + d(v, w) differ by at most 2δ ≥ 0. Hyperbolicity can be viewed as a measure of how close a graph is to a tree metrically; the smaller the hyperbolicity of a graph, the closer it is metrically to a tree. A graph G is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G there exists the smallest Helly graph H(G) into which G isometrically embeds (H(G) is called the injective hull of G) and the hyperbolicity of H(G) is equal to the hyperbolicity of G. Motivated by this, we investigate structural properties of Helly graphs that govern their hyperbolicity and identify three isometric subgraphs of the King-grid as structural obstructions to a small hyperbolicity in Helly graphs.

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 αi-metric (i ∈ N ) if it satisfies the following αi-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,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 α0-metric graphs are exactly ptolemaic graphs, and that chordal graphs and distance-hereditary graphs are αi-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 αi-metric graph can be computed in total linear time. Our strongest results are obtained for α1-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 (α1, ∆)-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 αi-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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