In graph theory, a partial cube is a graph that is an isometric subgraph of a hypercube. 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. Such a labeling is called a Hamming labeling; it represents an isometric embedding of the partial cube into a hypercube.

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.

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:

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. 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.

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. 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 ${\displaystyle O(nm)}$; 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.