On the geodetic Radon number of grids

It is NP-hard to determine the Radon number of graphs in the geodetic convexity. However, for certain classes of graphs, this well-known convexity parameter can be determined efficiently. In this paper, we focus on geodetic convexity spaces built upon d -dimensional grids, which are the Cartesian products of d paths. After revisiting a result of Eckhoff concerning the Radon number of R d in the convexity defined by Manhattan distance, we present a series of theoretical findings that disclose some very nice combinatorial aspects of the problem for grids. We also give closed expressions for the Radon number of the product of P 2 ’s and the product of P 3 ’s, as well as computer-aided results covering the Radon number of all possible Cartesian products of d paths for d ≤ 9 .