Lattice graph

A lattice graph, mesh graph, or grid graph, is a graph whose drawing, embedded in some Euclidean space Rn, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. A lattice graph, mesh graph, or grid graph, is a graph whose drawing, embedded in some Euclidean space Rn, forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in 'an 8×8 square grid'. The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs. 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 1,..., 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. A square grid graph is a Cartesian product of graphs, namely, of two path graphs with n − 1 {displaystyle n-1} and m − 1 {displaystyle m-1} edges. 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, which is easily verified by the fact that one can color the vertices in a checkerboard fashion. 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. Every planar graph H is a minor of the h×h-grid, where h = 2 | V ( H ) | + 4 | E ( H ) | {displaystyle h=2|V(H)|+4|E(H)|} . 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.