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Graphs and Cubes
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- Graphs and cubes.
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Graphs and Algorithms. Graphs and algorithms. Groups, graphs, and bases. For instance, r itself will have a label that is all zero bits, its neighbors will have labels with a single 1-bit, etc. Then the Hamming distance between any two labels is the distance between the two vertices in the tree, so this labeling shows that T is a partial cube.
Every hypercube graph is itself a partial cube, which can be labeled with all the different bitstrings of length equal to the dimension of the hypercube. Many of the theorems about partial cubes are based directly or indirectly upon a certain binary relation defined on the edges of the graph. This relation is reflexive and symmetric , but in general it is not transitive.
An embedding of a partial cube onto a hypercube of this dimension is unique, up to symmetries of the hypercube. Every hypercube and therefore every partial cube can be embedded isometrically into an integer lattice.
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The lattice dimension of a graph is the minimum dimension of an integer lattice into which the graph can be isometrically embedded. The lattice dimension may be significantly smaller than the isometric dimension; for instance, for a tree it is half the number of leaves in the tree rounded up to the nearest integer.
The lattice dimension of any graph, and a lattice embedding of minimum dimension, may be found in polynomial time by an algorithm based on maximum matching in an auxiliary graph.
Other types of dimension of partial cubes have also been defined, based on embeddings into more specialized structures. Isometric embeddings of graphs into hypercubes have an important application in chemical graph theory. A benzenoid graph is a graph consisting of all vertices and edges lying on and in the interior of a cycle in a hexagonal lattice.
Such graphs are the molecular graphs of the benzenoid hydrocarbons , a large class of organic molecules. Every such graph is a partial cube.