An upper bound on the P3-Radon number

The generalization of classical results about convex sets in Rn 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 P3-convexity on graphs. A set R of vertices of a graph G is P3-convex if no vertex in V (G) \ R has two neighbours in R. The P3-convex hull of a set of vertices is the smallest P3-convex set containing it. The P3-Radon number r(G) of a graph G is the smallest integer r such that every set R of r vertices of G has a partition R = R1∪R2 such that the P3-convex hulls of R1 and R2 intersect. We prove that r(G) ≤ 3 (n(G) + 1) + 1 for every connected graph G and characterize all extremal graphs. © 2012 Elsevier B.V. All rights reserved.