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Square grid graph Triangular grid graph. In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , 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.
In graph theory, a branch of mathematics, Halin's grid theorem states that the infinite graphs with thick ends are exactly the graphs containing subdivisions of the hexagonal tiling of the plane. [1] It was published by Rudolf Halin ( 1965 ), and is a precursor to the work of Robertson and Seymour linking treewidth to large grid minors , which ...
3. A square grid graph is a lattice graph defined from points in the plane with integer coordinates connected by unit-length edges. stable A stable set is a synonym for an independent set. star A star is a tree with one internal vertex; equivalently, it is a complete bipartite graph K 1,n for some n ≥ 2. The special case of a star with three ...
A connected graph may have a disconnected spanning forest, such as the forest with no edges, in which each vertex forms a single-vertex tree. [8] [9] A few graph theory authors define a spanning forest to be a maximal acyclic subgraph of the given graph, or equivalently a subgraph consisting of a spanning tree in each connected component of the ...
A planar separator for a grid graph. Consider a grid graph with rows and columns; the number of vertices equals .For instance, in the illustration, =, =, and = =.If is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if is odd, there is a single central column, and otherwise there are two columns equally close to the center.
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. [1] If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. [2] For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3.
The planar graphs do not have bounded treewidth, because the n × n grid graph is a planar graph with treewidth exactly n. Therefore, if F is a minor-closed graph family with bounded treewidth, it cannot include all planar graphs.
William Thurston () describes a test for determining whether a simply-connected region, formed as the union of unit squares in the plane, has a domino tiling.He forms an undirected graph that has as its vertices the points (x,y,z) in the three-dimensional integer lattice, where each such point is connected to four neighbors: if x + y is even, then (x,y,z) is connected to (x + 1,y,z + 1), (x ...