# Daisy cubes: a characterization and a generalization

## TL;DR

This paper characterizes daisy cubes, a class of isometric subgraphs of hypercubes, provides an expansion-based characterization, and introduces algorithms for embedding and generalizations.

## Contribution

It offers a new characterization of daisy cubes via an expansion procedure and presents algorithms for embedding graphs into hypercubes.

## Key findings

- Characterization of daisy cubes through expansion procedure
- Algorithm for embedding graphs into hypercubes in O(mn) time
- Introduction of daisy graphs as a generalization of daisy cubes

## Abstract

Daisy cubes are a recently introduced class of isometric subgraphs of hypercubes $Q_n$. They are induced with intervals between chosen vertices of $Q_n$ and the vertex $0^n\in V(Q_n)$. In this paper we characterize daisy cubes in terms of an expansion procedure thus answering an open problem proposed by Klav\v{z}ar and Mollard, 2018, in the introductory paper of daisy cubes \cite{KlaMol-18}. To obtain such a characterization several interesting properties of daisy cubes are presented. For a given graph $G$ isomorphic to a daisy cube, but without the corresponding embedding into a hypercube, we present an algorithm which finds a proper embedding of $G$ into a hypercube in $O(mn)$ time. Finally, daisy graphs of a rooted graph are introduced and shown to be a generalization of daisy cubes.