Complexity aspects of the triangle path convexity

In a triangle path  v1,…,vtv1,…,vt of a graph GG no edges exist joining vertices vivi and vjvj such that |j−i|>2|j−i|>2. A set S⊆V(G)S⊆V(G) is convex in the triangle path convexity of     GG, or tt-convex  , if the vertices of every triangle path joining two vertices of SS are in SS. The cardinality of a maximum proper tt-convex subset of V(G)V(G) is the tt-convexity number of     GG and the cardinality of a minimum set of vertices whose tt-convex hull is V(G)V(G) is the tt-hull number of     GG. We solve the fundamental complexity problems on the triangle path convexity. Among them, we present polynomial-time algorithms for determining the tt-convexity number and the tt-hull number of a general graph.