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

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.

💡 Research Summary

The paper addresses a conjecture proposed by R. Hickingbotham concerning “k‑ghost‑edges” in graphs of bounded treewidth. For a connected graph G with treewidth tw(G) ≤ k, a non‑edge xy (i.e., xy ∈ E(Gᶜ)) is called a k‑ghost‑edge if, in every tree‑decomposition (T, ℬ) of G whose width does not exceed k, the two vertices x and y appear together in at least one bag. Hickingbotham proved that if there are at least k + 1 internally vertex‑disjoint (x, y)‑paths then xy must be a k‑ghost‑edge. He then conjectured the converse: if there are at most k such internally disjoint paths, then xy cannot be a k‑ghost‑edge.

The authors disprove this conjecture by constructing an explicit counterexample. They define a graph G (illustrated in Figure 1 of the paper) that has treewidth exactly 4. The construction is carefully designed: five “branch sets” form a K₅‑minor, guaranteeing tw(G) ≥ 4. By splitting G into two isomorphic subgraphs H₁ and H₂ (each containing x, y, and a set of auxiliary vertices) and providing explicit width‑4 tree‑decompositions for each (shown in Figure 2), they combine the two decompositions via a new bag {x, y, d₂, d₄} to obtain a global tree‑decomposition of G of width 4. Hence tw(G)=4.

The crucial part of the argument shows that despite there being exactly four internally vertex‑disjoint (x, y)‑paths (which is ≤ k for k = 4), the pair xy behaves as a 4‑ghost‑edge. The authors first prove a structural claim: any vertex set S with |S| ≤ 4 that separates x from y must be precisely the set A = {a₁, a₂, a₃, a₄}. This follows from the existence of four independent (x, y)‑paths and the 2‑connected nature of the components after removing {x, y, d₂, d₄}.

Assuming, for contradiction, that xy is not a 4‑ghost‑edge, they consider an optimal width‑4 tree‑decomposition (T, ℬ) where the subtrees Tₓ and Tᵧ (the sets of bags containing x and y, respectively) are disjoint. Let s ∈ Tₓ and t ∈ Tᵧ be the closest vertices in these subtrees. By the earlier claim, the bags Bₛ and Bₜ must be A ∪ {x} and A ∪ {y}, respectively. Since each bag has size five, they are maximal under the width constraint. The authors then examine the structure of the induced subgraph H consisting of Bₛ together with all vertices adjacent to Bₛ. Using the connectivity condition of tree‑decompositions (the “running intersection” property), they show that Bₛ cannot be a leaf of the decomposition tree (otherwise a bag would be properly contained in another, contradicting the assumption that no bag is a subset of another). If Bₛ is not a leaf, then removing Bₛ would disconnect H, which contradicts the observed connectivity of H \ Bₛ. Both possibilities lead to contradictions, forcing the conclusion that Tₓ and Tᵧ must intersect; consequently, x and y appear together in some bag of every width‑4 decomposition, i.e., xy is a 4‑ghost‑edge.

Thus the paper establishes that the converse direction of Hickingbotham’s conjecture fails: a pair of vertices can be a k‑ghost‑edge even when the number of internally vertex‑disjoint (x, y)‑paths does not exceed k. The counterexample also illustrates how minor‑based arguments (the K₅‑minor guaranteeing treewidth) and careful manipulation of tree‑decompositions can be combined to produce pathological cases. The result narrows the understanding of ghost‑edges, indicating that the existence of at most k disjoint paths is not sufficient to rule out ghost‑edge behavior. It opens new questions about precise characterizations of k‑ghost‑edges, potential additional structural conditions that might guarantee the conjectured direction, and the broader relationship between graph minors, treewidth, and the “shape” of tree‑decomposition trees.