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The queen’s 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 (Qn )a ndi(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 nd dominating sets. New values and bounds: (A) (Qn )= dn=2e is shown for 17 values of n (in particular, the set of values for which the conjecture (Q4k+1 )=2 k + 1 is known to hold is extended tok 32); (B) i(Qn )= dn=2e is shown for 11 values of n, including 5 of those from (A); (C) One or both of (Qn )a ndi(Qn) is shown to lie in fdn=2e, dn=2e +1 g 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 (Qn )a ndi(Qn) is either known, or known to have one of two values. Also, the general bounds (Qn) 69n=133 + O(1) and i(Qn) 61n=111 + O(1) are established.

Values of Domination Numbers of the Queen's Graph