King's graph

In graph theory, a king's graph is a graph that represents all legal moves of the king chess piece on a chessboard where each vertex represents a square on a chessboard and each edge is a legal move. More specifically, an n × m king's graph is a king's graph of an n × m chessboard. It is the map graph formed from the squares of a chessboard by making a vertex for each square and an edge for each two squares that share an edge or a corner. It can also be constructed as the strong product of two path graphs.

For a n × m king's graph the total number of vertices is n m and the number of edges is 4 n m − 3 ( n + m ) + 2 . For a square n × n king's graph, the total number of vertices is n 2 and the total number of edges is ( 2 n − 2 ) ( 2 n − 1 ) .

The neighbourhood of a vertex in the king's graph corresponds to the Moore neighborhood for cellular automata. 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.