Perfect Domination Ratios of Archimedean Lattices

Abstract

An Archimedean lattice is an infinite graph constructed from a vertex-transitive tiling of the plane by regular polygons. A dominating set of vertices is a perfect dominating set if every vertex that is not in the set is dominated exactly once. The perfect domination ratio is the minimum proportion of vertices in a perfect dominating set. Seven of the eleven Archimedean lattices can be efficiently dominated, which easily determines their perfect domination ratios. The perfect domination ratios are determined for the four Archimedean lattices that can not be efficiently dominated.