Values of Domination Numbers of the Queen's Graph

Abstract

The queen's graph $Q_{n}$ has the squares of the $n \times n$ chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let $\gamma (Q_{n})$ and $i(Q_{n})$ be the minimum sizes of a dominating set and an independent dominating set of $Q_{n}$, respectively. Recent results, the Parallelogram Law, and a search algorithm adapted from Knuth are used to find dominating sets. New values and bounds:

(A) $\gamma (Q_n) = \lceil n/2 \rceil$ is shown for 17 values of $n$ (in particular, the set of values for which the conjecture $\gamma (Q_{4k+1}) = 2k + 1$ is known to hold is extended to $k \leq 32$);

(B) $i(Q_n) = \lceil n/2 \rceil$ is shown for 11 values of $n$, including 5 of those from (A);

(C) One or both of $\gamma (Q_n)$ and $i(Q_n)$ is shown to lie in $\{ \lceil n/2 \rceil $, $\lceil n/2 \rceil + 1 \}$ for 85 values of $n$ distinct from those in (A) and (B).

Combined with previously published work, these results imply that for $n \leq 120$, each of $\gamma (Q_n)$ and $i(Q_n)$ is either known, or known to have one of two values.

Also, the general bounds $\gamma (Q_n) \leq 69n/133 + O(1)$ and $i(Q_n) \leq 61n/111 + O(1)$ are established.

Comment added August 25th 2003.

Corrigendum added October 5th 2017.