Complexity aspects of the triangle path convexity

A path P = v1, . . . , vt is a triangle path (respectively, monophonic path) of G if no edges exist joining vertices vi and vj of P such that |j − i| > 2; (respectively, |j − i| > 1). A set of vertices S is convex in the triangle path convexity (respectively, monophonic convexity) of G if the vertices of every triangle path (respectively, monophonic path) joining two vertices of S are in S. The cardinality of a maximum proper convex set of G is the convexity number of G and the cardinality of a minimum set of vertices whose convex hull is V (G) is the hull number of G. Our main results are polynomial time algorithms for determining the convexity number and the hull number of a graph in the triangle path convexity.