A299029 Triangle read by rows: Independent domination number for rectangular queens graph Q(n,m), 1 <= n <= m.

1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 1, 2, 2, 3, 3, 4, 1, 2, 3, 3, 4, 4, 4, 1, 2, 3, 4, 4, 4, 5, 5, 1, 2, 3, 4, 4, 4, 5, 5, 5, 1, 2, 3, 4, 4, 4, 5, 5, 5, 5, 1, 2, 3, 4, 4, 5, 5, 6, 5, 5, 5, 1, 2, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 1, 2, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 1, 2, 3, 4, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8

Offset: 1

Author

Sandor Bozoki, Feb 01 2018

Keywords

• nonn
• tabl

Comments

The queens graph Q(n X m) has the squares of the n X m chessboard as its vertices; two squares are adjacent if they are both in the same row, column, or diagonal of the board. A set D of squares of Q(n X m) is a dominating set for Q(n X m) if every square of Q(n X m) is either in D or adjacent to a square in D. The minimum size of an independent dominating set of Q(n X m) is the independent domination number, denoted by i(Q(n X m)).

Less formally, i(Q(n X m)) is the number of independent queens that are necessary and sufficient to all squares of the n X m chessboard be occupied or attacked.

Chessboards 8 X 11 and 18 X 11 are of special interest, because they cannot be dominated by 5 and 8 independent queens, respectively, although the larger boards 9 X 11, 10 X 11, 11 X 11 and 18 X 12 are. It is open how many such counterexamples of this kind of monotonicity exist.

Examples

			Table begins
  m\n| 1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18
  ---+-----------------------------------------------------
   1 | 1
   2 | 1  1
   3 | 1  1  1
   4 | 1  2  2  3
   5 | 1  2  2  3  3
   6 | 1  2  2  3  3  4
   7 | 1  2  3  3  4  4  4
   8 | 1  2  3  4  4  4  5  5
   9 | 1  2  3  4  4  4  5  5  5
  10 | 1  2  3  4  4  4  5  5  5  5
  11 | 1  2  3  4  4  5  5  6  5  5  5
  12 | 1  2  3  4  4  5  5  6  6  6  6  7
  13 | 1  2  3  4  5  5  6  6  6  7  7  7  7
  14 | 1  2  3  4  5  6  6  6  6  7  7  8  8  8
  15 | 1  2  3  4  5  6  6  7  7  7  7  8  8  9  9
  16 | 1  2  3  4  5  6  6  7  7  7  8  8  8  9  9  9
  17 | 1  2  3  4  5  6  7  7  7  8  8  8  9  9  9  9  9
  18 | 1  2  3  4  5  6  7  7  8  8  9  8  9  9  9 10 10 10
		

Links

• Sandor Bozoki, First 18 rows of the triangle, formatted as a simple linear sequence n, a(n) for n = 1..171
• S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, Domination of the rectangular queens graph, arXiv:1606.02060 [math.CO], 2016.
• S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, Domination of the rectangular queens graph, 2016.
• Eric Weisstein's World of Mathematics, Queen Graph
• Eric Weisstein's World of Mathematics, Queens Problem

Crossrefs

Diagonal elements are in A075324: Independent domination number for queens graph Q(n).

Cf. A274138: Domination number for rectangular queens graph Q(n,m).

Cf. A279404: Independent domination number for queens graph on an n X n toroidal board.