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%I #17 Mar 05 2018 17:57:45
%S 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,
%T 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,
%U 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
%N Triangle read by rows: Independent domination number for rectangular queens graph Q(n,m), 1 <= n <= m.
%C 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)).
%C 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.
%C 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.
%H Sandor Bozoki, <a href="/A299029/b299029.txt">First 18 rows of the triangle, formatted as a simple linear sequence n, a(n) for n = 1..171</a>
%H S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, <a href="http://arxiv.org/abs/1606.02060">Domination of the rectangular queens graph</a>, arXiv:1606.02060 [math.CO], 2016.
%H S. Bozóki, P. Gál, I. Marosi, W. D. Weakley, <a href="http://www.sztaki.mta.hu/~bozoki/queens/">Domination of the rectangular queens graph</a>, 2016.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QueenGraph.html">Queen Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/QueensProblem.html">Queens Problem</a>
%e Table begins
%e m\n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
%e ---+-----------------------------------------------------
%e 1 | 1
%e 2 | 1 1
%e 3 | 1 1 1
%e 4 | 1 2 2 3
%e 5 | 1 2 2 3 3
%e 6 | 1 2 2 3 3 4
%e 7 | 1 2 3 3 4 4 4
%e 8 | 1 2 3 4 4 4 5 5
%e 9 | 1 2 3 4 4 4 5 5 5
%e 10 | 1 2 3 4 4 4 5 5 5 5
%e 11 | 1 2 3 4 4 5 5 6 5 5 5
%e 12 | 1 2 3 4 4 5 5 6 6 6 6 7
%e 13 | 1 2 3 4 5 5 6 6 6 7 7 7 7
%e 14 | 1 2 3 4 5 6 6 6 6 7 7 8 8 8
%e 15 | 1 2 3 4 5 6 6 7 7 7 7 8 8 9 9
%e 16 | 1 2 3 4 5 6 6 7 7 7 8 8 8 9 9 9
%e 17 | 1 2 3 4 5 6 7 7 7 8 8 8 9 9 9 9 9
%e 18 | 1 2 3 4 5 6 7 7 8 8 9 8 9 9 9 10 10 10
%Y Diagonal elements are in A075324: Independent domination number for queens graph Q(n).
%Y Cf. A274138: Domination number for rectangular queens graph Q(n,m).
%Y Cf. A279404: Independent domination number for queens graph on an n X n toroidal board.
%K nonn,tabl
%O 1,8
%A _Sandor Bozoki_, Feb 01 2018
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