A counterexample to Hickingbotham's conjecture about $k$-ghost-edges

TL;DR

This paper disproves Hickingbotham's conjecture by providing a counterexample, showing that the relationship between disjoint paths and $k$-ghost-edges is more complex than previously thought.

Contribution

The paper presents a counterexample to Hickingbotham's conjecture, challenging the assumed link between disjoint paths and $k$-ghost-edges in graphs with bounded treewidth.

Findings

Counterexample disproves the conjecture

Disjoint paths do not determine $k$-ghost-edges

Complexity of $k$-ghost-edges in relation to disjoint paths

Abstract

Fix $k\in \mathbb{N}$ and let $G$ be a connected graph with $tw(G)\leq k$. We say that $xy\in E(G^c)$ is a {\em $k$-ghost-edge} of $G$ if for every tree decomposition $(T,\cB)$ of $G$ with width at most $k$, the set $\{x,y\}$ is contained in a bag of $(T,\cB)$. Although a $k$-ghost-edge of $G$ is not an edge of $G$, but it behaves like real edges with respect to tree decomposition of $G$ with width at most $k$. For any graph $G$ with treewidth $k$ and $xy\in E(G^c)$, when there are at least $k+1$ internally vertex disjoint $(x,y)$-paths, Hickingbotham proved that $xy$ is a $k$-ghost-edge of $G$; while when there are at most $k$ internally vertex disjoint $(x,y)$-paths, he conjectured that it is not a $k$-ghost-edge of $G$. In this paper, we prove that this conjecture is wrong.