Inapproximability results for graph convexity parameters

In this paper, we prove several inapproximability results on the P 3 -convexity and the geodesic convexity in graphs. We prove that determining the P 3 -hull number and the geodetic hull number are APX-hard problems. We prove that the Caratheodory number, the Radon number and the convexity number of both convexities are O ( n 1 - e ) -inapproximable in polynomial time for every e 0 , unless P = NP . We also prove that the interval numbers of both convexities are W 2 -hard and O ( log ? n ) -inapproximable in polynomial time, unless P = NP . Moreover, these results hold for bipartite graphs in the P 3 -convexity.

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