Partial cube
[1] In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that the distance between any two vertices in the partial cube is the same as the distance between those vertices in the hypercube.
Equivalently, a partial cube is a graph whose vertices can be labeled with bit strings of equal length in such a way that the distance between two vertices in the graph is equal to the Hamming distance between their labels.
The graphs that admit such embeddings were characterized by Djoković (1973) and Winkler (1984), and were later named partial cubes.
A separate line of research on the same structures, in the terminology of families of sets rather than of hypercube labelings of graphs, was followed by Kuzmin & Ovchinnikov (1975) and Falmagne & Doignon (1997), among others.
More complex examples include the following: 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, first described by Djoković (1973) and given an equivalent definition in terms of distances by Winkler (1984), is denoted by
Winkler showed that a connected graph is a partial cube if and only if it is bipartite and the relation
A Hamming labeling may be obtained by assigning one bit of each label to each of the equivalence classes of the Djoković–Winkler relation; in one of the two connected subgraphs separated by an equivalence class of edges, all of the vertices have a 0 in that position of their labels, and in the other connected subgraph all of the vertices have a 1 in the same position.
[9] Given a partial cube, it is straightforward to construct the equivalence classes of the Djoković–Winkler relation by doing a breadth first search from each vertex, in total time
-time recognition algorithm speeds this up by using bit-level parallelism to perform multiple breadth first searches in a single pass through the graph, and then applies a separate algorithm to verify that the result of this computation is a valid partial cube labeling.
[10] Every hypercube and therefore every partial cube can be embedded isometrically into an integer lattice.
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.
[11] Other types of dimension of partial cubes have also been defined, based on embeddings into more specialized structures.
A Hamming labeling of such a graph can be used to compute the Wiener index of the corresponding molecule, which can then be used to predict certain of its chemical properties.
