In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space \mathbb{R}^n, 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 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, 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 H is a minor of the h × h grid, where . Grid graphs are fundamental objects in the theory of graph minors because of the grid exclusion theorem. They play a major role in bidimensionality theory. A triangular grid graph is a graph that corresponds to a triangular grid.

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Concepts associés (13)
Apex graph
In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K_5 or K_3,3, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.
Graphe chemin
In the mathematical field of graph theory, a path graph (or linear graph) is a graph whose vertices can be listed in the order v_1, v_2, ..., v_n such that the edges are {v_i, v_i+1} where i = 1, 2, ..., n − 1. Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph.
Graphe distance-unité
En mathématiques, plus particulièrement en théorie des graphes, un graphe distance-unité est un graphe s'obtenant à partir d'un ensemble de points du plan euclidien en reliant par une arête toutes les paires de points étant à une distance de 1. Les arêtes peuvent se croiser si bien qu'un graphe distance-unité n'est pas nécessairement un graphe planaire. S'il n'y a pas de croisement entre les arêtes, alors le graphe est qualifié de graphe allumette.
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