On the construction of tree decompositions of hypercubes

TL;DR

This paper reviews existing knowledge on tree decompositions of hypercubes and introduces new explicit constructions for both known and unknown cases, clarifying the field.

Contribution

It provides a comprehensive overview and new explicit methods for constructing tree decompositions of hypercubes, including previously unresolved cases.

Findings

Summarizes known results on hypercube tree decompositions

Introduces new explicit constructions for various cases

Clarifies the landscape of hypercube decomposition methods

Abstract

There are different concepts regarding to tree decomposition of a graph $G$. For the Hypercube $Q_n$, these concepts have been shown to have many applications. But some diverse papers on this subject make it difficult to follow what is precisely known. In this note first we will mention some known results on the tree decomposition of hypercubes and then introduce new explicit constructions for the previously known and unknown cases.