Values of Domination Numbers of the Queen's Graph

• Published in: The Electronic journal of combinatorics 2001-03, Vol.8 (1)
• Main Authors: Östergård, Patric R. J., Weakley, William D.
• Format: Article
• Language: English
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/1573

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.