Efficient and Non-efficient Domination of Z-stacked Archimedean Lattices

Introduction

On a graph, a vertex v dominates vertex ?′ if v=?′ or ? is adjacent to ?′. An efficient dominating set of a graph is a subset of the graph’s vertices D such that every vertex in the graph is dominated by exactly one vertex in D. We investigate the existence of efficient dominating sets for the stacked versions of each of the 11 Archimedean lattices. We prove that 7 of the Z-stacked Archimedean lattices have efficient dominating sets, while the other 4 do not.