Map 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 interior-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. 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.

Map graphs can be represented combinatorially as the "half-squares of planar bipartite graphs". That is, let G = (U,V,E) be a planar bipartite graph, with bipartition (U,V). The square of G is another graph on the same vertex set, in which two vertices are adjacent in the square when they are at most two steps apart in G. The half-square or bipartite half is the induced subgraph of one side of the bipartition (say V): its vertex set is V and it has an edge between each two vertices in V that are two steps apart in G. The half-square is a map graph. It can be represented geometrically by finding a planar embedding of G, and expanding each vertex of V and its adjacent edges into a star-shaped region, so that these regions touch at the vertices of U. Conversely, every map graph can be represented as a half-square in this way.

...
Wikipedia