Inapproximability results and bounds for the Helly and Radon numbers of a graph

Abstract Let C be a convexity on a set X and denote the convex hull of S ⊆ X in C by H ( S ) . The Helly number (Radon number) of C is the minimum integer k such that, for every S ⊆ X with at least k + 1 elements, it holds ⋂ v ∈ S H ( S ∖ { v } ) ≠ ∅ (there is a bipartition of S into sets S 1 and S 2 with H ( S 1 ) ∩ H ( S 2 ) ≠ ∅ ). In this work, we show that there is no approximation algorithm for the Helly or the Radon number of a graph G of order n in the geodetic convexity to within a factor n 1 − e for any e > 0 , unless P = NP, even if G is bipartite. Furthermore, we present upper bounds for both parameters in the geodetic convexity of bipartite graphs and characterize the families of graphs achieving the bound for the Helly number.