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A legal placement of Queens is any placement of Queens on an order N chessboard in which any two attacking Queens can be separated by a Pawn. The Queens independence separation number is the minimum number of Pawns which can be placed on an N × N board to result in a separated board on which a maximum of m independent Queens can be placed. We prove...
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Citations
... In [Cha09], given a chess piece C and a graph parameter π, the π-separation number s C (π, n, p) for C is defined as the minimum number of pawns for which some placement of those pawns on an n × n board will produce a board whose C graph has π = p. Following that pattern, we define the rooks diameter-separation number s R (diam, n, d) to be the minimum number of pawns for which some placement of those pawns on an n × n board will produce a board whose rooks graph has diameter d. ...
We define the queens (resp., rooks) diameter-separation number to be the minimum number of pawns for which some placement of those pawns on an n × n board produces a board with a queens graph (resp., rooks graph) with a desired diameter d . We determine these numbers for some small values of d .
... The domatic number of is the maximum cardinality of a domatic partition of and it is detoned by ( ). The domatic number was introduced by Cockayne and Hedetniemi [4]. Proof: Case(i) Suppose = 3 . ...
The independent domatic queen number of a graph Q n is the maximum number of pairwise disjoint minimum independent queen dominating sets of P n and it is denoted by i d ( Q n ) while maximum independent domatic queen number is denoted by I d ( Q n ). We discuss about in this paper the independent (maximum independent) domatic number of queen graph Q n on n × n chess board.
... The Queens separation number s Q (M, N ) is the minimum number of pawns that can separate some legal placement of M queens on an N ×N board [4]. The Queens independence separation number s Q (β, l, M × N ) is the minimum number of pawns that can be placed on an M × N board to result in a separated board on which a maximum of l independent queens can be placed [3]. ...
... Using this, Chatham et al.,in [4] defined the board obtained from the addition of one or more pawns to be a separated board. He further proved the separation of N + k queens by k pawns for large enough N in [3] and then introduced some domination-related parameters to separation problems with other chess pieces. ...
... Therefore, when k = 1, we add 2 columns, one for the pawn and the other for the extra queen. For the other values of k, we follow the same process as mentioned above for the case when M To prove our next theorem we use the following result from Section.4 in [3] which explains the restriction of pawns location on an N × N board. ...
The famous eight queens problem with non-attacking queens placement on an 8 × 8 chessboard was posed in the year 1848. The Queens separation problem is the legal placement of the fewest number of pawns with the maximum number of independent queens placed on an N × N board which results in a separated board. Here a legal placement is defined as the separation of attacking queens by pawns. Using this concept, the current study extends the Queens separation problem onto the rectangular board M × N (M < N) to result in a separated board with the maximum number of independent queens. The research work here first shows that M + k queens are separated with 1 pawn and continues to prove that k pawns are required to separate M + k queens. Then it focuses on finding the symmetric solutions to the M + k Queens separation problem.
... A lot of work has been done in various domination parameters on different chessboard graphs starting with square boards [4], and then this work has been extended to queens separation problems. Chatham et al.,in [3] found the queens separation on square chessboard, and in [2] he determined the separation problem of various other chess pieces with independence and domination parameters on square chessboards. ...
The Independence Number (�) is the number of vertices in a maximum independent domi-
nating set. The Independent Dominating Set D is a subset of vertices V where D is a dominating set
with non-adjacent vertices. The aim of this paper is to bring the concept of separation problem on square
hexagonal chessboard (i.e., a square board with hexagonal cells). The queens independence separation
number, sQ(�, n + k, n) is the placement of maximum of n + k non-attacking queens on an n × n board
using minimum number of pawns k. Here we are interested in finding the queens and bishops indepen-
dence separation numbers on a square hexagonal board which we denote by sQH(�, �(QH) + k, n) and
sBH(�, �(BH) + k, n) respectively.
... Several papers in this area and their extensions were carried out, where Zhao [8] in 1998 extended it further by showing that more than n independent queens can be placed on an n × n board if enough blocking pieces, such as pawns, are placed between queens. Chatham et al. further extended the work of Zhao, and brought the concept of separated chessboard graphs with various domination parameters on an n × n board starting with independence domination in [2,3]. Chatham et al. defined the separation problem as the legal placement of minimum number of pawns with the maximum number of independent chess pieces chosen on an n × n board which results in a separated board. ...
... For example, Zhao determined that three pawns were necessary and sufficient to allow six independent queens on a 5 × 5 board [3]. Chatham et al. further explored the concept of placing pawns on a chessboard as obstacles in order to obtain "separated" chessboard graphs with altered graph parameters [4][5][6][7], with an emphasis on aspects of the "n + k queens problem" of finding arrangements of n + k independent queens and k pawns on an n × n chessboard. Only a few results were found regarding independence-separation on rook's and bishop's graphs. ...
... We use the notation of [2,6]. Let π be a graph parameter defined on the rook's graph, the queen's graph, or the bishop's graph, and a a non-negative integer. ...
... The number of -1s in such a matrix is k, so the number of pawns in the corresponding chessboard arrangement is also k. So we can place n + k independent rooks and k pawns on an n × n board if and only if 0 In [6] it was shown that s B (β, 2n, n) = 1 for odd n and an open question was posed relating to the values of s B (β, 2n, n) for even n. We answer that question in Theorem 7 and determine bounds for independence-separation numbers on the bishop's graph which show s B (β, 2n − 2 + d + k, n) = k in many cases, with d ≡ n (mod 2). ...
For a chessboard graph and a given graph parameter π, a π separation number is the minimum number of pawns for which some arrangement of those pawns on the board will produce a board where π has some desired value. We extend previous results on independence and domination separation. We also consider separation of other domination-related parameters.
... Zhao proved that we need 3 pawns to put 6 queens on a 5 × 5 board [6]. For each k > 0, we can place n + k mutually non-attacking queens on an n × n board with k pawns, if n is sufficiently large [2,Theorem 11]. Figure 2: Nine mutually non-attacking queens with one pawn on a standard chessboard. ...
The classic n-queens problem asks for placements of just n mutually non-attacking queens on an n × n board. By adding enough pawns, we can arrange to fill roughly one-quarter of the board with mutually non-attacking queens. How many pawns do we need? We discuss that question for square boards as well as rectangular m × n boards.
... In this paper we consider special solutions to the "N + k queens problem", which calls for placing N + k queens 'Q' and k pawns 'P' on an N × N board so that no two queens attack each other. It was conjectured in [8] and proved in [6] that for each k ≥ 0, there is a number N (k) depending on k such that if N > N (k) then the N +k queens problem has at least one solution. In [7] algorithms that count the number of solutions to the N + k queens problem for various values of N and k were presented and compared. ...
... Given are both the total number of solutions and the total number of fundamental solutions, the latter referring to solutions that cannot be transformed into one another by rotations and/or reflections. [6,7] found solutions to the general N + k queens problem by enumerating all possible configurations of k pawns on the N × N chessboard, subject to certain constraints, and for each such configuration using an implementation of Knuth's Algorithm X to place the queens. Algorithm X is a backtracking algorithm that uses a data structure that Knuth calls Dancing Links (DLX) [11]. ...
... Algorithm X is a backtracking algorithm that uses a data structure that Knuth calls Dancing Links (DLX) [11]. Following [6] and [7], we refer to this algorithm for solving the N + k queens problem as DLX. DLX was selected based on performance comparisons to a standard backtracking algorithm using array storage for both the N queens and N + k queens problems. ...
Chessboard separation problems are modifications to classic chessboard problems, such as the N queens problem, in which obstacles are placed on the chessboard. The N + k queens problem requires placements of k pawns and N + k mutually non-attacking queens on an N -by-N chessboard. Here we examine centrosymmetric (half-turn symmetric) and doubly centrosymmetric (quarter-turn symmetric) solutions to the N + k queens problem. We also consider solutions in which the queens have a different type of symmetry than the pawns have.
... The "N + k Queens Problem" is the problem of placing N + k queens and k pawns on an N × N board so that no two queens attack each other. It was conjectured in [4] and proven in [3] that for each k ≥ 0, for large enough N , the N + k Queens Problem has at least one solution. In this paper we consider algorithms that count the number of solutions to the N + k Queens Problem for various values of N and k. ...
... Two different exhaustive search methods, recursive backtracking and dancing links, for solving the N + k Queens Problem were examined in [3]. We provide improved versions of these two methods in Section 2.1. ...
... In [3], solutions to the N + k Queens Problem based on traditional recursive backtracking and dancing links were considered and compared, but not optimized. Backtracking was implemented as a standard backtracking algorithm, placing a queen in each row and proceeding. ...
Abstract Chessboard separation problems are modiflcations to classic chess- board problems, such as the N Queens Problem, in which obstacles are placed on the chessboard. This paper focuses on a variation known as the N + k Queens Problem, in which k Pawns and N + k mutually non-attacking Queens are to be placed on an N-by-N chess- board. Results are presented from performance studies examining the e‐ciency of sequential and parallel programs that count the number of solutions to the N + k Queens Problem using traditional back- tracking and dancing links. The use of Stochastic Local Search for determining existence of solutions is also presented. In addition, pre- liminary results are given for a similar problem, the N +k Amazons.
... For other k, according to [4,Theorem 11], if N > max{87 + k, 25k}, then there is a solution to the N + k queens problem. The proof also involves taking known solutions to the M queens problem for some M < N , and adding extra rows, columns, queens, and pawns in systematic ways to obtain patterns that can be verified as being N + k queens problem solutions. ...
... The above result, combined with computer searches, tells us that the N + 2 queens problem has solutions for all N ≥ 7, and that the N + 3 queens problem has solutions for all N ≥ 8 [4, p. 8]. This evidence, and the results in Table 1, leads us to believe that the bounds given by [4,Theorem 11] can be greatly decreased. In any event, the following question is open. ...
... For more information about the N queens problem, see [7], [15], and [16]. For more about the N + k queens problem and related problems, see [6], [4], and [5] (Preprints are available online at [3]). For those interested in alternating sign matrices, we suggest [1], [2], and [12]. ...
Given a regular chessboard, can you place eight queens on it, so that no two queens attack each other? More generally, given a square chessboard with N rows and N columns, can you place N queens on it, so that no two queens attack each other? Figure 1. A solution to the eight queens problem. This puzzle, known as the N queens problem, is old, and famous, and has an ex-tensive history. Here we present a recently formulated elaboration, which we call the N + k queens problem. We describe some of what is known about the N + k queens problem, prove a few new results, and propose some open questions.
