A note on the weakly convex and convex domination numbers of a torus

The distance  dG(u,v)dG(u,v) between two vertices uu and vv in a connected graph GG is the length of the shortest (u,v)(u,v) path in GG. A (u,v)(u,v) path of length dG(u,v)dG(u,v) is called a (u,v)(u,v)-geodesic  . A set X⊆VX⊆V is called weakly convex   in GG if for every two vertices a,b∈Xa,b∈X, exists an (a,b)(a,b)-geodesic, all of whose vertices belong to XX. A set XX is convex   in GG if for all a,b∈Xa,b∈X all vertices from every (a,b)(a,b)-geodesic belong to XX. The weakly convex domination number   of a graph GG is the minimum cardinality of a weakly convex dominating set of GG, while the convex domination number   of a graph GG is the minimum cardinality of a convex dominating set of GG. In this paper we consider weakly convex and convex domination numbers of tori.