On the Convexity of Paths of Length Two in Undirected Graphs

A subset S ⊆ V ( G ) of a graph G is p 2 convex when v , w ∈ S and z ∈ V ( G ) imply z ∈ S , whenever v, z, w is a path of G. If S = V ( G ) then S is a p 2 set of G. The size of the smallest p 2 set of G is the p 2 number of G, while the size of the largest proper p 2 convex set is the p 2 convexity number of G. On the other hand, for any given subset S of V ( G ) , the smallest convex set S h containing S is the p 2 hull set of S. If S h = V ( G ) then S h is a p 2 hull set of G. The size of the smallest p 2 hull set is the p 2 hull number of G. In this work, we prove the NP-hardness of the determination of p 2 number and p 2 convexity number of a graph, and describe polynomial time algorithms for trees, cographs and classes of grids.