Properties of queens graphs and the irredundance number of \(Q_7\) (Q2712529)

The vertices of the queens graph \(Q_{n}\) are the \(n^{2}\) squares of the chessboard, and two squares are adjacent if they are collinear. This graph has received much attention in the literature recently because of the well-known century-old problem of determining the domination number \(\gamma (Q_{n})\) which remains unsolved. It is known that a domination set of a graph is minimal if and only if it is also irredundant. NEWLINENEWLINENEWLINEIn this paper some results concerning common neighbours of vertex subsets and irredundance in the queens graph \(Q_{n}\) are proved. Also it is shown that the lower irredundance number of \(Q_{7}\) is equal to four.