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

Abstract It is well known that the Cartesian product of two paths, the grid P m □ P n , 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 = 2 k + 1 . In this paper we prove that every grid P m □ P n 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.