Perfect Domination and Cube-Sphere Tilings of Zn
Fuentes Castilla, Luis R.
Dejter, Italo J. (Consejero)
We study efficient dominating sets in the integer lattices as well as some re- lated topics. In particular, we study the existence of lattice-like tilings of the n- dimensional integer lattice by the union of a cartesian product of paths and an isolated vertex and we present two types of these lattice-like tilings. We also study the existence of lattice-like tilings of the n-dimensional integer lattice by generalized Lee spheres around cubes of at least two different dimensions. While this was already known for n = 2, a perfect dominating set for n = 3 is constructed whose induced components are squares and isolated vertices. In generalizing this, an extension of the notion of generalized Lee sphere in a graph-theoretical context is given to one of cube-sphere. A lattice-like cube-sphere tiling of the n-dimensional integer lattice by the connected union of two generalized Lee spheres of radius 1 around (n−1)-cubes and two cube-spheres of radius n − 2 around isolated vertices is given. Finally, we prove that there are not non-lattice-like tiling in the 3-dimensional integer lattice with squares.