# Partial cubes: structures, characterizations, and constructions

## TL;DR

This paper explores the structures, characterizations, and construction methods of partial cubes, providing new insights and proofs, and analyzing how various operations affect their dimensions.

## Contribution

It introduces new characterizations of partial cubes, offers alternative proofs of existing results, and studies the effects of operations like Cartesian product and pasting on their dimensions.

## Key findings

- New characterizations of partial cubes and bipartite graphs.
- Proofs of known results using new methods.
- Calculations of isometric and lattice dimensions after operations.

## Abstract

Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi\'{c}'s and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.