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| | The author is recognized by the title of Numbers Lint. Body building is one of the issues I love most. I am a meter reader but I plan on altering it. Years in the past we moved to Puerto Rico and my family enjoys it.<br><br>my web blog; over the counter std test - [http://phpfoxdev.azurewebsites.net/index.php?do=/profile-44231/info/ similar web-site], |
| | name = King's graph
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| | image = [[Image:King's graph.svg|180px]]
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| | image_caption = 8x8 King's graph
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| | vertices = ''nm''
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| | edges = 4''nm''-3(''n''+''m'')+2
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| | chromatic_number =
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| | girth =
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| In [[graph theory]], a '''king's graph''' is a [[Graph (mathematics)|graph]] that represents all legal moves of the [[king (chess)|king]] [[chess]] [[chess piece|piece]] on a [[chessboard]] where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an <math>n \times m</math> king's graph is a king's graph of an <math>n \times m</math> chessboard.<ref>{{citation
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| | last = Chang | first = Gerard J.
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| | editor1-last = Du | editor1-first = Ding-Zhu
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| | editor2-last = Pardalos | editor2-first = Panos M.
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| | contribution = Algorithmic aspects of domination in graphs
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| | location = Boston, MA
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| | mr = 1665419
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| | pages = 339–405
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| | publisher = Kluwer Acad. Publ.
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| | title = Handbook of combinatorial optimization, Vol. 3
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| | year = 1998}}. Chang defines the king's graph on [http://books.google.com/books?id=w0rmms0_hMMC&pg=PA341 p. 341].</ref>
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| For a <math>n \times m</math> king's graph the total number of vertices is simply <math>n m</math>. For a <math>n \times n</math> king's graph the total number of vertices is simply <math>n^2</math> and the total number of edges is <math>(2n-2)(2n-1)</math>.<ref>{{SloanesRef|A002943}}</ref>
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| The [[Neighbourhood (graph theory)|neighbourhood of a vertex]] in the king's graph corresponds to the [[Moore neighborhood]] for cellular automata.<ref>{{citation
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| | last = Smith | first = Alvy Ray | authorlink = Alvy Ray Smith
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| | contribution = Two-dimensional formal languages and pattern recognition by cellular automata
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| | doi = 10.1109/SWAT.1971.29
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| | pages = 144–152
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| | title = 12th Annual Symposium on Switching and Automata Theory
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| | year = 1971}}.</ref>
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| A generalization of the king's graph, called a '''kinggraph''', is formed from a [[squaregraph]] (a planar graph in which each bounded face is a quadrilateral and each interior vertex has at least four neighbors) by adding the two diagonals of every quadrilateral face of the squaregraph.<ref>{{citation
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| | last1 = Chepoi | first1 = Victor
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| | last2 = Dragan | first2 = Feodor
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| | last3 = Vaxès | first3 = Yann
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| | contribution = Center and diameter problems in plane triangulations and quadrangulations
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| | pages = 346–355
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| | title = Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '02)
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| | year = 2002}}.</ref>
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| ==References==
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| {{reflist}}
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| ==See also==
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| * [[Knight's graph]]
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| * [[Rook's graph]]
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| * [[Lattice graph]]
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| [[Category:Mathematical chess problems]]
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| [[Category:Parametric families of graphs]]
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The author is recognized by the title of Numbers Lint. Body building is one of the issues I love most. I am a meter reader but I plan on altering it. Years in the past we moved to Puerto Rico and my family enjoys it.
my web blog; over the counter std test - similar web-site,