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A299029 Triangle read by rows: Independent domination number for rectangular queens graph Q(n,m), 1 <= n <= m. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

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.

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

EXAMPLE

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

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.

Sequence in context: A097028 A289442 A092331 * A200747 A328481 A089293

Adjacent sequences:  A299026 A299027 A299028 * A299030 A299031 A299032

KEYWORD

nonn,tabl

AUTHOR

Sandor Bozoki, Feb 01 2018

STATUS

approved

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Last modified July 27 18:27 EDT 2021. Contains 346308 sequences. (Running on oeis4.)