Every grid has an independent [1, 2]-set

Abstract

It is well known that the Cartesian product of two paths, the grid Pm□Pn,m≤n, has an independent dominating set such that every vertex not in the set has exactly one neighbor in it, if and only if m=n=4 or m=2,n=2k+1. In this paper we prove that every grid Pm□Pn has an independent dominating set such that every vertex not in the set has at most two neighbors in it, and we also calculate the minimum cardinality of such sets.