Values of domination numbers of the queen's graph

Abstract

The queen's graph Qn has the squares of the n x 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. Recent results, the Parallelogram Law, and a search algorithm adapted from Knuth are used to find dominating sets. New values and bounds: (A) γ(Qn) = ⌈n/2⌉ is shown for 17 values of n (in particular, the set of values for which the conjecture γ(Q4k+1) = 2k + 1 is known to hold is extended to k ≤ 32); (B) i(Qn) = ⌈n/2⌉ is shown for 11 values of n, including 5 of those from (A); (C) One or both of γ(Qn) and i(Qn) is shown to lie in {⌈n/2⌉, ⌈n/2⌉ + 1} for 85 values of n distinct from those in (A) and (B). Combined with previously published work, these results imply that for n ≤ 120, each of γ(Qn) and i(Qn) is either known, or known to have one of two values. Also, the general bounds γ(Qn) ≤ 69n/133 + O(1) and i(Qn) ≤ 61n/111 + O(1) are established.