Lattice graph

Ingraph theory,alattice graph,mesh graph,orgrid graphis agraphwhosedrawing,embeddedin someEuclidean space⁠$\mathbb {R} ^{n}$⁠,forms aregular tiling.This implies that thegroupofbijective transformationsthat send the graph to itself is alattice in the group-theoretical sense.

Square gridgraph

Triangular gridgraph

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 alattice,mesh,orgrid.Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid".

The termlattice graphhas also been given in the literature to various other kinds of graphs with some regular structure, such as theCartesian productof a number ofcomplete graphs.

Square grid graph

A common type of lattice graph (known under different names, such asgrid graphorsquare grid graph) is the graph whose vertices correspond to the points in the plane withintegercoordinates,x-coordinatesbeing in the range1, ..., n,y-coordinatesbeing in the range1, ..., m,and two vertices being connected by an edge whenever the corresponding points are at distance 1. In other words, it is theunit distance graphfor the integer points in a rectangle with sides parallel to the axes.

Properties

A square grid graph is aCartesian product of graphs,namely, of twopath graphswithn− 1 andm− 1 edges.Since a path graph is amedian graph,the latter fact implies that the square grid graph is also a median graph. All square grid graphs arebipartite,which is easily verified by the fact that one can color the vertices in a checkerboard fashion.

A path graph is a grid graph on the$1\times n$grid. A$2\times 2$grid graph is a4-cycle.

Everyplanar graphHis aminorof theh × hgrid, where$h=2|V(H)|+4|E(H)|$.

Grid graphs are fundamental objects in the theory of graph minors because of thegrid exclusion theorem.They play a major role inbidimensionality theory.

Other kinds

Atriangular grid graphis a graph that corresponds to a triangular grid.

AHanan gridgraph 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.

Therook's graph(the graph that represents all legal moves of therookchess pieceon achessboard) is also sometimes called thelattice graph,although this graph is different from the lattice graph described here because all points in one row or column are adjacent. The valid moves of thefairy chess piecethewazirform a square lattice graph.

See also

• Lattice path
• Pick's theorem
• Integer triangles in a 2D lattice
• Regular graph