Abstract
A ( strongly) Helly graph* is a connected graph for which any finite (resp. finite or infinite) family of pairwise non-disjoint balls has a non-empty intersection. Strongly Helly graphs are important objects of the category of simple graphs with contractions (i.e., maps that preserve or contract the edges). They are the absolute retracts and the injective objects with respect to the isometries. Many properties about (strongly) Helly graphs are known, in particular results concerning the geodesic convexity and fixed point properties or more precisely invariant simplex properties. We will survey several of these results.
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