Complexity aspects of the computation of the rank of a graph

We consider the P-3-convexity on simple undirected graphs, in which a set of vertices S is convex if no vertex outside S has two or more neighbors in S. The convex hull H (S) of a set S is the smallest convex set containing S as a subset. A set S is a convexly independent set if v is not an element of H (S\{v}) for all v in S. The rank rk(G) of a graph is the size of the largest convexly independent set. In this paper we consider the complexity of determining rk(G). We show that the problem is NP-complete even for split or bipartite graphs with small diameter. We also show how to determine rk(G) in polynomial time for the well structured classes of graphs of trees and threshold graphs. Finally, we give a tight upper bound for rk(G), which in turn gives a tight upper bound for the Radon number as byproduct, which is the same obtained before by Henning, Rautenbach and Schafer. Additionally, we briefly show that the problem is NP-complete also in the monophonic convexity.