Map graph

A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron)
The Four Corners of the USA. Even though these four states meet at a point, rather than sharing a boundary of nonzero length, they form adjacent vertices in the corresponding map graph.
The king's graph, the map graph of squares of the chessboard. A chess king can move between any two adjacent vertices of this graph.

In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.[1] Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.

  1. ^ Chen, Zhi-Zhong; Grigni, Michelangelo; Papadimitriou, Christos H. (2002), "Map graphs", Journal of the ACM, 49 (2): 127–138, arXiv:cs/9910013, doi:10.1145/506147.506148, MR 2147819, S2CID 2657838.