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This chapter contains a survey of results that have been obtained on domination, independent domination, irredundance, and total domination in chessboard graphs, such as queens, kings, bishops, knights, and rooks graphs.

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... According to Haynes, Hedetniemi, and Henning [15], more than 4000 papers on the subject were published by the year 2020. Domination problems originated in the 19th century in chess [17,16]. A placement of chess pieces on a chessboard is called dominating if each free square of the chessboard is under attack by at least one piece. ...

... G 1×1 (z) = z G 2×2 (z) = 6 z 2 + 4 z 3 + z 4 G 3×3 (z) = 48 z 3 + 117 z 4 + 126 z 5 + 84 z 6 + 36 z 7 + 9 z 8 + z 9 G 4×4 (z) = 40 z 4 + 560 z 5 + 2 736 z 6 + 6 800 z 7 + 10 310 z 8 + 10 560 z 9 + 7 832 z 10 + 4 352 z 11 + 1 820 z 12 + 560 z 13 + 120 z 14 + 16 z 15 + z 16 G 5×5 (z) = 10 z 5 + 200 z 6 + 3 050 z 7 + 31 525 z 8 + 188 700 z 9 + 677 690 z 10 + 1 610 700 z 11 + 2 740 775 z 12 + 3 527 075 z 13 + 3 562 700 z 14 + 2 895 610 z 15 + 1 923 600 z 16 ...

We present an algorithm to compute the domination polynomial of the $m \times n$ grid, cylinder, and torus graphs and the king graph. The time complexity of the algorithm is $O(m^2n^2 \lambda^{2m})$ for the torus and $O(m^3n^2\lambda^m)$ for the other graphs, where $\lambda = 1+\sqrt{2}$. The space complexity is $O(mn\lambda^m)$ for all of these graphs. We use this algorithm to compute domination polynomials for graphs up to size $24\times 24$ and the total number of dominating sets for even larger graphs. This allows us to give precise estimates of the asymptotic growth rates of the number of dominating sets. We also extend several sequences in the Online Encyclopedia of Integer Sequences.

... The 2 × k queen graph, Q 2k . From Hedetniemi and Hedetniemi (2021) we have (Q 2k ) = k 2 for k ≥ 1. 2. The 2 × k rook graph, K 2 K k . From Burcroff (2021) we have (K 2 K k ) = k for k ≥ 1. 3. The k×k rook graph, K k K k . ...

... Graphs based on chess boards and pieces have been widely studied for domination problems including upper domination (e.g. see Hedetniemi and Hedetniemi 2021), and provide non-trivial and structurally interesting instances. Grid graphs have also been extensively considered for domination problems (Alanko et al. 2011;Dorfling and Henning 2006;Gonçalves et al. 2011;Gravier 2002); the result showing that the domination number of grid graphs separates into 23 individual cases is a wonderful example of the surprising complexity of these instances. ...

We consider Upper Domination, the problem of finding the minimal dominating set of maximum cardinality. Very few exact algorithms have been described for solving Upper Domination. In particular, no binary programming formulations for Upper Domination have been described in literature, although such formulations have proved quite successful for other kinds of domination problems. We introduce two such binary programming formulations, and show that both can be improved with the addition of extra constraints which reduce the number of feasible solutions. We compare the performance of the formulations on various kinds of graphs, and demonstrate that (a) the additional constraints improve the performance of both formulations, and (b) the first formulation outperforms the second in most cases, although the second performs better for very sparse graphs. Also included is a short proof that the upper domination number of any generalized Petersen graph P(n, k) is equal to n.

... We summarise the most noteworthy of these results for the variants of domination considered in this paper. For upper domination, results are known for various graphs based on chessboards [4,11,18,39,40]. For total domination, results are known for some grid graphs G(n, m) (for n ≤ 6) [15,21], Knödel graphs [27], and various graphs from chemistry [26]. ...

We consider the flower snarks, a widely studied infinite family of 3--regular graphs. For the Flower snark Jn on 4n vertices, it is trivial to show that the domination number of Jn is equal to n. However, results are more difficult to determine for variants of domination. The Roman domination, weakly convex domination, and convex domination numbers have been determined for flower snarks in previous works. We add to this literature by determining the independent domination, 2-domination, total domination, connected domination, upper domination, secure Domination and weak Roman domination numbers for flower snarks.

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S. The minimum size of a kTDS is called the k-tuple total dominating number and it is denoted by γ×k,t(G). We give a constructive proof of a general formula for γ×3,t (Kn Km ).

Given a (symmetrically-moving) piece from a chesslike game, such as shogi, and an n×n board, we can form a graph with a vertex for each square and an edge between two vertices if the piece can move from one vertex to the other. We consider two pieces from shogi: the dragon king, which moves like a rook and king from chess, and the dragon horse, which moves like a bishop and rook from chess. We show that the independence number for the dragon kings graph equals the independence number for the queens graph. We show that the (independent) domination number of the dragon kings graph is n − 2 for 4 ≤ n ≤ 6 and n − 3 for n ≥ 7. For the dragon horses graph, we show that the independence number is 2n − 3 for n ≥ 5, the domination number is at most n−1 for n ≥ 4, and the independent domination number is at most n for n ≥ 5.

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.

We consider the independence, domination and independent domination numbers of graphs obtained from the moves of queens on chessboards drawn on the torus, and determine exact values for each of these parameters in infinitely many cases.

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 that N + k Queens can be separated by k Pawns for large enough N and provide some results on the number of fundamental solutions to this problem. We also introduce separation relative to other domination-related parameters for Queens, Rooks, and Bishops.

In Martin Gardner's October 1976 Mathematical Games column in Scientific American, he posed the following problem: "What is the smallest number of [queens] you can put on an [n x n chessboard] such that no [queen] can be added without creating three in a row, a column, or, except in the case when n is congruent to 3 modulo 4, in which case one less may suffice." We use the Combinatorial Nullstellensatz to prove that this number is at least n. A second, more elementary proof is also offered in the case that n is even.

The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

An elementary treatment of a class of solutions to the n-queens problem leads to a proof of Fermat's theorem on primes which are sums of two squares.

In this paper, we count the number of non-attacking bishop and king positions on the regular and cylindrical m x n (where m = 1, 2 and 3) chessboards. This is accomplished through the use of scientific computing, recurrence relations, generating functions and closed-form formulas.

For more than 250 years combinatorial problems on chessboards have been studied and published in numerous books on recreational mathematics. Two problems of this type include the problem of finding a placement of n non-attacking queens on an n×n chessboard and the problem of determining the minimum number of queens which are necessary to cover every square of an n×n chesboard. Within the past five years a surge of interest in chessboard problems has occurred among a group of a dozen or so graph theorists and computer scientists. This paper surveys recent developments and mentions a large number of open problems.

We prove results concerning common neighbours of vertex subsets and irredundance in the queens graph Qn. We also establish that the lower irredundance number of Q7 is equal to four.

We describe various computing techniques for tackling chessboard domination problems and apply these to the determination of domination and irredundance numbers for queens' and kings' graphs. In particular we confirm that γ(Q17) = γ(Q18) = 9, and show that γ(Q14) = 8, γ(Q15) γ(Q16) = 9, γ(Q19) = 10, i(Q18) = 10, 10 ≤ i(Q19) ≤ 11, ir(Qn) = γ(Qn) for 1 ≤ n ≤ 13, IR(Q9) = Γ(Q9) = 13, IR(Q10) = Γ(Q10) = 15, γ(Q4k+1) = 2k + 1 for k = 16, 18, 20 and 21, i(Q22) ≤ 12, IR(K8) = 17, IR(K9) = 25, IR(K10) = 27, and IR(K11) = 36. We calculate the number of non-isomorphic minimum dominating and independent dominating sets in the queens' graph Qn for n ≤ 15 and n ≤ 18 respectively.

The queen’s graph Q n has the squares of the n×n chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let γ(Q n ) and i(Q n ) be the minimum sizes of a dominating set and an independent dominating set of Q n , respectively. P. H. Spencer has proved γ(Q n )≥(n-1)/2. We show γ(Q 4k+1 )≥2k+1 for all k, and that i(Q 4k+1 )=γ(Q 4k+1 )=2k+1 for k≤6 and k=8.

The domination number γ(Qn) of the queens graph Qn is the minimum number of queens required to cover every square on an n×n chessboard. We discuss properties of dominating sets of Q4k+3 of cardinality 2k+1 and show that such sets do not exist for 3 ≤ k ≤ 7. This result yields the new domination numbers γ(Q19) = 10 and γ (Q31) = 16.

Denote the n × n toroidal queens graph by Qtn. We show that γ(Qt3k) = k + 2 when k ≡ 0, 3, 4, 6, 8, 9 (mod 12). This completes the proof that γ(Qt3k) = 2k - β(Qtk) for all positive integers k.

Computing techniques are described which have resulted in the establishment of new results for the queens domination problem. In particular it is shown that the minimum cardinalities of independent sets of dominating queens for chessboards of size 14, 15, and 16 are 8, 9, and 9 respectively, and that the minimum cardinalities of sets of dominating queens for chessboards of size 29, 41, 45, and 57 are 15, 21,23 and 29 respectively. As a by-product the numbers of non-isomorphic ways of covering a chessboard of size n with k independent queens for 1 :; n :; 15 and 1 :; k :; 8, as well as the case n = 16, k = 8, are computed.

An addendum to [P.A. Burchett, D. Lane, and J.A. Lachniet, “k-tuple and k-domination on the rook’s graph and other results,” Congr. Numerantium 199, 187–204; Zbl 1211.05102)] is given.

Considered in this paper are k-domination and k-tuple domination on the rook’s graph. For a graph G=(V,E) a set S is a k-tuple dominating set if every vertex in V is dominated at least k times. Given the same graph G, a set S is a k-dominating set if every vertex in V∖S is dominated at least k times. Upper and lower bounds are given for both γ k (R n ) and γ ×k (R n ). These bounds are used to solve the k-tuple domination number on the rook’s graph for the cases where k is odd, and when k=2 and k=2n-2. In a similar fashion, the provided bounds are also used to solve the k-domination number on the rook’s graph when n≥k 2 -2k 4 and k is even, and the cases where k=2, k=3, k=4, k=6, and k≥2n-3. Lastly, 2-tuple domination, also known as double domination, is considered on the queen’s graph and solved on the rook’s graph. The bound 8 11(n-1)≤dd(Q n )≤n is arrived at along with dd(Q n ) values for 2≤n≤10.

We define moves for king, queen, rook, bishop, and knight on a triangular honeycomb chessboard. Domination and independence numbers on this board for each piece are analyzed.

We describe a simple computing technique for solving independence and domination problems on rectangular chessboards. It rests upon relational modeling and uses the BDD-based specific purpose computer algebra system RelView for the evaluation of the relation-algebraic expressions that specify the problems’ solutions and the visualization of the computed results. The technique described in the paper is very flexible and especially appropriate for experimentation. It can easily be applied to other chessboard problems.

We show that the minimum number of queens required to cover the n×n chessboard is at most .

Abstract Total dominating sets and connected dominating sets in Queens graphs are two topics which haven’t been studied enough in computationaltheory. We try to find an upper bound and a lower bound for the total domination number,γt(Qn) and connected domination number,γc(Qn) of the Queens graph Qn. Our work is mostly based on the concept of a p-cover, which has been previously used to find the domination number γ(Qn) and independent domination number i(Qn), for most values of n≤100. We introduce a search algorithm to find total and connected dominating sets for Qn,for n≤30, which achieve our lower bound in several cases.

A set SV is a dominating set of a graph G = (V;E) if each vertex in V is either in S or is adjacent to a vertex in S. A vertex is said to dominate itself and all its neighbors. The domination number (G) is the minimum cardinality of a dominating set of G. In terms of a chess board problem, let Xn be the graph for chess piece X on the square of side n. Thus, (Xn) is the domination number for chess piece X on the square of side n. In 1964, Yaglom and Yaglom established that (Kn) = � n+2 3 � 2 : This extends to (Km;n) = � m+2 3 �� n+2 3 � for the rectangular board. A set SV is a total dominating set of a graph G = (V;E) if each vertex in V is adjacent to a vertex in S. A vertex is said to dominate its neighbors but not itself. The total domination number t (G) is the minimum cardinality of a total dominating set of G. In 1995, Garnick and Nieuwejaar conducted an analysis of the total domination numbers for the king's graph on the mn board. In this paper we note an error in one portion of their analysis and provide a correct general upperbound for t (Km;n): Furthermore, we state improved upperbounds for t (Kn).

A set D of vertices of a finite, undirected graph G = (V, E) is a total dominating set if every vertex of V is adjacent to some vertex of D. In this paper we initiate the study of total dominating sets in graphs and, in particular, obtain results concerning the total domination number of G (the smallest number of vertices in a total dominating set) and the total domatic number of G (the largest order of a partition of G into total dominating sets).

An 1874 article by J. W. L. Glaisher asserted that the eight queens problem of recreational mathematics originated in 1850 with Franz Nauck proposing it to Gauss, who then gave the complete solution. In fact the problem was first proposed two years earlier by Max Bezzel, proposed again by Nauck in a newspaper Gauss happened to read, and only partially solved by Gauss in a casual attempt. Glaisher had access to an accurate account of the history in German but perhaps could not read the language well; the error subsequently spread through the recreational mathematics literature.RésuméEn 1874, J. W. L. Glaisher affirmait dans un article que le problème des huit reines des mathématiques récréatives avait été énoncé pour la première fois par Franz Nauck en 1850 lorsque ce dernier le proposa à Gauss qui en aurait donné alors une solution complète. En fait, Max Bezzel avait déjà proposé ce problème deux années auparavant. Il fut à nouveau énoncé par Nauck dans un journal que Gauss vint à lire. Celui-ci le solutionna partiellement dans une tentative informelle. Glaisher avait accès à une description juste de ces faits en allemand, mais peut-être ne pouvait-il pas lire adéquatement cette langue. Il s'ensuivit que l'erreur se répendit dans toute la littérature sur les mathématiques récréatives.

The minimum number of queens which can be placed on ann n chessboard so that all other squares are dominated by at least one queen but no queen covers another, is shown to be less than 0.705n + 2.305.

A vertex x in a subset X of vertices of an undirected graph is redundant if its closed neighbourhood is contained in the union of closed neighbourhoods of vertices of X−{x}. In the context of a communications network, this means that any vertex which may receive communications from X may also be informed from X−{x}. The lower and upper irredundance numbers ir(G) and IR(G) are respectively the minimum and maximum cardinalities taken over all maximal sets of vertices having no redundancies. The domination number γ(G) and upper domination number Γ(G) are respectively the minimum and maximum cardinalities taken over all minimal dominating sets of G. The independent domination number i(G) and the independence number β(G) are respectively the minimum and maximum cardinalities taken over all maximal independent sets of vertices of G.A variety of inequalities involving these quantities are established and sufficient conditions for the equality of the three upper parameters are given. In particular a conjecture of Hoyler and Cockayne [9], namely , is proved.