Lattice graph

The terms lattice graph, mesh graph, or grid graph refer to a number of categories of graphs whose drawing corresponds to some grid/mesh/lattice, i.e., its vertices correspond to the nodes of the mesh and its edges correspond to the ties between the nodes.

Square grid graph

A common type of a lattice graph (known under different names, such as square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 0,..., n, y-coordinates being in the range 1,...m, and two vertices are connected by an edge whenever the corresponding points are at distance 1. In other words, it is a unit distance graph for the described point set.^[1]

Properties

A square grid graph is a Cartesian product of graphs, namely, of two path graphs with n and m edges.^[1] Since a path graph is a median graph, the latter fact implies that the square grid graph is also a median graph. All grid graphs are bipartite.

A path graph may also be considered to be a grid graph on the grid n times 1. A 2x2 grid graph is a 4-cycle.^[1]

Other kinds

A triangular grid graph is a graph that corresponds to a triangular grid.

A Hanan grid graph for a finite set of points in the plane is produced by the grid obtained by intersections of all vertical and horizontal lines through each point of the set.

The rook's graph (the graph that represents all legal moves of the rook chess piece on a chessboard) is also sometimes called the lattice graph.

References

1. ^ ^{a b c} CRC Concise Encyclopedia of Mathematics, by Eric W. Weisstein, article "Grid graph"; Weisstein, Eric W., "Grid graph" from MathWorld.