On tree decompositions whose trees are subgraphs

TL;DR

This paper investigates a special class of graphs called $k$-ghost-free graphs and disproves a conjecture that such graphs always admit a tree decomposition with the tree as a subgraph of the original graph for all $k \\geq 3$.

Contribution

The paper provides a counterexample to Hickingbotham's conjecture, showing it does not hold for all $k \\geq 3$, thus advancing understanding of tree decompositions.

Findings

Hickingbotham's conjecture is false for all $k \\geq 3$.

Counterexamples exist for the conjecture.

The structure of $k$-ghost-free graphs is more complex than previously thought.

Abstract

Fix $k \in \mathbb{N}$ and let $G$ be a connected graph with treewidth at most $k$. We say that $xy \notin E(G)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T, \cB)$ of $G$ with width at most $k$, both $x$ and $y$ are contained in a bag of $(T, \cB)$. Moreover, if $G$ does not contain any $k$-ghost-edges, then $G$ is {\em $k$-ghost-free}. Hickingbotham proposed a conjecture that every connected $k$-ghost-free graph $G$ has a tree decomposition $(T, \cB)$ with width at most $k$ such that $T$ is a subgraph of $G$. In this paper, we prove that Hickingbotham's conjecture is false for all $k\geq3$.