Partial cube

In

of the partial cube into a hypercube.

History

Firsov (1965) was the first to study isometric embeddings of graphs into hypercubes. 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.^[2]

Examples

Every tree is a partial cube. For, suppose that a tree T has m edges, and number these edges (arbitrarily) from 0 to m – 1. Choose a root vertex r for the tree, arbitrarily, and label each vertex v with a string of m bits that has a 1 in position i whenever edge i lies on the path from r to v in T. 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.

More complex examples include the following:

• Consider the graph whose vertex labels consist of all possible (2n + 1)-digit bitstrings that have either n or n + 1 nonzero bits, where two vertices are adjacent whenever their labels differ by a single bit. This labeling defines an embedding of these graphs into a hypercube (the graph of all bitstrings of a given length, with the same adjacency-condition) that turns out to be distance-preserving. The resulting graph is a bipartite Kneser graph; the graph formed in this way with n = 2 has 20 vertices and 30 edges, and is called the Desargues graph.
• All median graphs are partial cubes.^[3] The trees and hypercube graphs are examples of median graphs. Since the median graphs include the squaregraphs, simplex graphs, and Fibonacci cubes, as well as the covering graphs of finite distributive lattices, these are all partial cubes.
• The
  hyperplane arrangement in Euclidean space of any number of dimensions, the graph that has a vertex for each cell of the arrangement and an edge for each two adjacent cells is a partial cube.^[4]
• A partial cube in which every vertex has exactly three neighbors is known as a cubic partial cube. Although several infinite families of cubic partial cubes are known, together with many other sporadic examples, the only known cubic partial cube that is not a planar graph is the Desargues graph.^[5]
• The underlying graph of any antimatroid, having a vertex for each set in the antimatroid and an edge for every two sets that differ by a single element, is always a partial cube.
• The Cartesian product of any finite set of partial cubes is another partial cube.^[6]
• A subdivision of a complete graph is a partial cube if and only if either every complete graph edge is subdivided into a two-edge path, or there is one complete graph vertex whose incident edges are all unsubdivided and all non-incident edges have been subdivided into even-length paths.^[7]

The Djoković–Winkler relation

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 ${\displaystyle \Theta }$. Two edges ${\displaystyle e=\{x,y\}}$ and ${\displaystyle f=\{u,v\}}$ are defined to be in the relation ${\displaystyle \Theta }$, written ${\displaystyle e{\mathrel {\Theta }}f}$, if ${\displaystyle d(x,u)+d(y,v)\not =d(x,v)+d(y,u)}$. This relation is reflexive and symmetric, but in general it is not transitive.

Winkler showed that a connected graph is a partial cube if and only if it is bipartite and the relation ${\displaystyle \Theta }$ is transitive.^[8] In this case, it forms an equivalence relation and each equivalence class separates two connected subgraphs of the graph from each other. 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.

Recognition

Partial cubes can be recognized, and a Hamming labeling constructed, in ${\displaystyle O(n^{2})}$ time, where ${\displaystyle n}$ is the number of vertices in the graph.

from each vertex, in total time ${\displaystyle O(nm)}$; the ${\displaystyle O(n^{2})}$-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.

Dimension

The isometric dimension of a partial cube is the minimum dimension of a hypercube onto which it may be isometrically embedded, and is equal to the number of equivalence classes of the Djoković–Winkler relation. For instance, the isometric dimension of an ${\displaystyle n}$-vertex tree is its number of edges, ${\displaystyle n-1}$. An embedding of a partial cube onto a hypercube of this dimension is unique, up to symmetries of the hypercube.^[10]

Every hypercube and therefore every partial cube can be embedded isometrically into an

Other types of dimension of partial cubes have also been defined, based on embeddings into more specialized structures.[12]

Application to chemical graph theory

Isometric embeddings of graphs into hypercubes have an important application in

A different molecular structure formed from carbon, the diamond cubic, also forms partial cube graphs.^[14]

Notes

References

• Beaudou, Laurent; Gravier, Sylvain; Meslem, Kahina (2008), "Isometric embeddings of subdivided complete graphs in the hypercube" (PDF), SIAM Journal on Discrete Mathematics, 22 (3): 1226–1238,
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• Hadlock, F.; Hoffman, F. (1978), "Manhattan trees", Utilitas Mathematica, 13: 55–67. As cited by Ovchinnikov (2011).
• Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons,
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• Klavžar, Sandi; Gutman, Ivan;
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• Kuzmin, V.; Ovchinnikov, S. (1975), "Geometry of preferences spaces I", Automation and Remote Control, 36: 2059–2063. As cited by Ovchinnikov (2011).
• Ovchinnikov, Sergei (2011), Graphs and Cubes, Universitext, Springer. See especially Chapter 5, "Partial Cubes", pp. 127–181.
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