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# Recreational Mathematics Magazine

2 Issues per year

Mathematical Citation Quotient (MCQ) 2016: 0.05

Open Access
Online
ISSN
2182-1976
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Volume 4, Issue 8

# Independence and domination on shogiboard graphs

Doug Chatham
Published Online: 2018-01-11 | DOI: https://doi.org/10.1515/rmm-2017-0018

## 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.

Keywords: shogi; n-queens problem; combinatorics

## References

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

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

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Published Online: 2018-01-11

Published in Print: 2017-12-20

Citation Information: Recreational Mathematics Magazine, Volume 4, Issue 8, Pages 25–37, ISSN (Online) 2182-1976,

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