Algorithmic and structural aspects of the -Radon number

The generalization of classical results about convex sets in [R.sup.n] to abstract convexity spaces, defined by sets of paths in graphs, leads to many challenging structural and algorithmic problems. Here we study the Radon number for the [P.sub.3]-convexity on graphs. [P.sub.3]-convexity has been proposed in connection with rumour and disease spreading processes in networks and the Radon number allows generalizations of Radon's classical convexity result. We establish hardness results and describe efficient algorithms for trees.