Independence and domination on shogiboard graphs

Open access

Abstract

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.

[1] Bell, J., Stevens, B. “A survey of known results and research areas for n-queens”, Discrete Math, 309, 1-31, 2009.

[2] Bodlaender, H., Duniho, F. “Shogi: Japanese chess”, 2017. http://www.chessvariants.com/shogi.html

[3] Chatham, D. “The maximum queens problem with pawns”, Recreational Mathematics Magazine, 3(6), 95-102, 2016.

[4] Haynes, T.W., Hedetniemi, S.T., Slater, P.J. Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.

[5] Haynes, T.W., Hedetniemi, S.T., Slater, P.J. (Eds.). Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998.

[6] Kosters, W. A. n-Queens bibliography, 2017. http://www.liacs.nl/home/kosters/nqueens/

[7] Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., Tack, G. “MiniZinc: Towards a standard CP modelling language”, in: C. Bessiere (editor), Proceedings of the 13th International Conference on Principles and Practice of Constraint Programming, volume 4741 of LNCS Springer, 529-543, 2007.

[8] Sloane, N.J.A. Sequence A002464 in The On-Line Encyclopedia of Integer Sequences, 2017. https://oeis.org

[9] Stuckey, P.J., Feydy, T., Schutt, A., Tack, G., Fischer, J. “The MiniZinc challenge 2008-2013”, AI Magazine, 35(2), 55-60, 2014.

[10] Watkins, J. J. Across the Board: The Mathematics of Chessboard Problems, Princeton University Press, 2004.

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Target Group

researchers in the fields of games and puzzles, problems, mathmagic, mathematics and arts, math and fun with algorithms

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