Efficient proper embedding of a daisy cube

TL;DR

This paper characterizes proper isometric embeddings of daisy cubes into hypercubes and provides a linear-time algorithm for finding such embeddings, enhancing understanding of their structure and embedding properties.

Contribution

It establishes a necessary and sufficient condition for proper isometric embeddings of daisy cubes and introduces an efficient linear-time algorithm for embedding them into hypercubes.

Findings

Proper embedding occurs if and only if the label 0^h is assigned to a minimal vertex.

The paper provides a linear-time algorithm for finding proper embeddings.

Characterizes the structure of daisy cubes and their embeddings into hypercubes.

Abstract

For a set $X$ of binary words of length $h$ the daisy cube $Q_h(X)$ is defined as the subgraph of the hypercube $Q_h$ induced by the set of all vertices on shortest paths that connect vertices of $X$ with the vertex $0 ^h$. A vertex in the intersection of all of these paths is a minimal vertex of a daisy cube. A graph $G$ isomorphic to a daisy cube admits several isometric embeddings into a hypercube. We show that an isometric embedding is proper if and only if the label $0 ^h$ is assigned to a minimal vertex of $G$. This result allows us to devise an algorithm which finds a proper embedding of a graph isomorphic to a daisy cube into a hypercube in linear time.