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graph theory In mathematics, graph theory is the study of '' graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conn ...
, a lattice graph, mesh graph, or grid graph is a graph whose
drawing Drawing is a form of visual art in which an artist uses instruments to mark paper or other two-dimensional surface. Drawing instruments include graphite pencils, pen and ink, various kinds of paints, inked brushes, colored pencils, crayo ...
, embedded in some
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, 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 In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\t ...
of a number of complete graphs.


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 An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coordinates, being in the range , being in the range , 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 In mathematics, particularly geometric graph theory, a unit distance graph is a graph formed from a collection of points in the Euclidean plane by connecting two points whenever the distance between them is exactly one. To distinguish these grap ...
for the described point set.


Properties

A square grid graph is a Cartesian product of graphs, namely, of two path graphs with ''n'' − 1 and ''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 square grid graphs are
bipartite Bipartite may refer to: * 2 (number) * Bipartite (theology), a philosophical term describing the human duality of body and soul * Bipartite graph, in mathematics, a graph in which the vertices are partitioned into two sets and every edge has an en ...
, 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 2 × 2 grid graph is a 4-cycle. Every
planar graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cro ...
''H'' is a minor of the ''h'' × ''h'' grid, where h = 2, V(H), + 4, E(H), .


Other kinds

A triangular grid graph is a graph that corresponds to a triangular grid. A
Hanan grid In geometry, the Hanan grid of a finite set of points in the plane is obtained by constructing vertical and horizontal lines through each point in . The main motivation for studying the Hanan grid stems from the fact that it is known to cont ...
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 A chess piece, or chessman, is a game piece that is placed on a chessboard to play the game of chess. It can be either white or black, and it can be one of six types: king, queen, rook, bishop, knight, or pawn. Chess sets generally come with ...
on a
chessboard A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the bo ...
) is also sometimes called the lattice graph, although this graph is strictly different than the lattice graph described in this article. The valid moves of fairy chess piece wazir form the square lattice graph.


See also

* Lattice path *
Pick's theorem In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick i ...
* Integer triangles in a 2D lattice *
Lattice (order) A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...
* Regular graph


References

{{reflist Planar graphs Graph families