Partial cubes with pre-hull number at most 1

TL;DR

This paper characterizes partial cubes among bipartite graphs using convexity properties and shows that those with pre-hull number at most 1 form a closed class under certain graph operations.

Contribution

It provides a new characterization of partial cubes with pre-hull number at most 1 based on convexity of attaching points and explores their closure properties.

Findings

Connected bipartite graphs with pre-hull number ≤ 1 are partial cubes.

The class of such partial cubes is closed under gated subgraphs, amalgams, and Cartesian products.

Abstract

We prove that a connected bipartite graph G is a partial cube if and only if the set of attaching points of any copoint of G is convex. A consequence of this result is that any connected bipartite graph with pre-hull number at most 1 is a partial cube. We show that the class of partial cubes with pre-hull number at most 1 is closed under gated subgraphs, gated amalgams and cartesian products.