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Domination in Isosceles Triangular Chessboard

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 احمد عبد علي عمران المعموري
19/12/2016 18:53:41
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One of the classical chessboard problems is placing minimum number of one type of pieces such that all
unoccupied positions are under attack. This problem is called the domination number problem of P. This
number is denoted by ?(P) and we denote the number of different methods for placing P pieces to obtain the
domination number of P by S(?(P)). Also we are interested in the domination number of two different types of
pieces by fixing a number  of one type P of pieces and determine the domination number ( ? , ) of
another type ?of pieces. Finally we compute (( ?, )).
In n square chessboard (see [4], [5] and [6]) ? is studied for rook "R", bishop "B" and king "K" .They proved
that ?(R) =  , ?(B) =  and ?(K) = 
 

.
In [3], JoeMaio and William proved that ?(R) = min {m, } for mxn Toroidal chessboards.
A saw-toothed chessboard, or STC for short, is a kind of chessboard whose boundary forms two staircases
from left down to right without any hole inside it.
In [2] Hon-Chan Chena, Ting-Yem Hob, determined the minimum number of rooks that can dominate all
squares of the STC.
Dietrich and Harborth [1] studied the triangular triangle board, i.e. the board in the shape of a triangle with
triangular cells. They defined the chess pieces in particular the rook which attacks in straight lines from a side
of the triangle to a side of the triangle, forming rhombuses, the bishop 1 which attacks from vertex to side, side
to vertex etc. in straight lines, forming diamonds, and for the bishop 2: the triangular triangle board can be 2-
coloured, whose cells are sharing an edge of different color, and the bishop 2 moves as bishop 1, but attacks
only cells of the same color.

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  • domination, Isosceles triangle chessboard, Kings, Bishops and Rooks