Abstract :
The queenʹs graph Qn has the squares of the n×n chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let γ(Qn) and i(Qn) be the minimum sizes of a dominating set and an independent dominating set of Qn, respectively. We show that if n≡1 (mod 4) and D is a d-element dominating set of Qn of a particular, commonly used kind, then for all k, γ(Qk)⩽(d+3)k/(n+2)+O(1). If also D is independent, then for all k, i(Qk)⩽(d+6)k/(n+2)+O(1). Other similar bounds are derived.
