# King's graph

King's graph
8x8 King's graph
Vertices nm
Edges 4nm-3(n+m)+2

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 ${\displaystyle n\times m}$ king's graph is a king's graph of an ${\displaystyle n\times m}$ chessboard.[1] 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.[2]

For a ${\displaystyle n\times m}$ king's graph the total number of vertices is ${\displaystyle nm}$ and the number of edges is ${\displaystyle 4nm-3(n+m)+2}$. For a square ${\displaystyle n\times n}$ king's graph, the total number of vertices is ${\displaystyle n^{2}}$ and the total number of edges is ${\displaystyle (2n-2)(2n-1)}$.[3]

The neighbourhood of a vertex in the king's graph corresponds to the Moore neighborhood for cellular automata.[4] 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.[5]

## References

1. ^ Chang, Gerard J. (1998), "Algorithmic aspects of domination in graphs", in Du, Ding-Zhu; Pardalos, Panos M., Handbook of combinatorial optimization, Vol. 3, Boston, MA: Kluwer Acad. Publ., pp. 339–405, MR 1665419. Chang defines the king's graph on p. 341.
2. ^ Berend, Daniel; Korach, Ephraim; Zucker, Shira (2005), "Two-anticoloring of planar and related graphs" (PDF), 2005 International Conference on Analysis of Algorithms, Discrete Mathematics & Theoretical Computer Science Proceedings, Nancy: Association for Discrete Mathematics & Theoretical Computer Science, pp. 335–341, MR 2193130.
3. ^ Sloane, N.J.A. (ed.). "Sequence A002943". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
4. ^ Smith, Alvy Ray (1971), "Two-dimensional formal languages and pattern recognition by cellular automata", 12th Annual Symposium on Switching and Automata Theory, pp. 144–152, doi:10.1109/SWAT.1971.29.
5. ^ Chepoi, Victor; Dragan, Feodor; Vaxès, Yann (2002), "Center and diameter problems in plane triangulations and quadrangulations", Proceedings of the Thirteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '02), pp. 346–355.