Inversion Diameter and Treewidth

In an oriented graph $\overrightarrow{G}$, the inversion of a subset $X$ of vertices is the operation that reverses the orientation of all arcs with both end-vertices in $X$. The inversion graph of a graph $G$, denoted by $\mathcal{I}(G)$, is the graph whose vertices are orientations of $G$ in which two orientations $\overrightarrow{G_1}$ and $\overrightarrow{G_2}$ are adjacent if and only if there is an inversion transforming $\overrightarrow{G_1}$ into $\overrightarrow{G_2}$. The inversion diameter of a graph $G$ is the diameter of its inversion graph $\mathcal{I}(G)$, denoted by $\diam(\mathcal{I}(G))$. Havet, Hörsch, and Rambaud~(2024) first proved that for $G$ of treewidth $k$, $\diam(\mathcal{I}(G)) \le 2k$, and that there are graphs of treewidth $k$ with inversion diameter $k+2$. In this paper, we construct graphs of treewidth $k$ with inversion diameter $2k$, which implies that the previous upper bound $\diam(\mathcal{I}(G)) \le 2k$ is tight. Moreover, for graphs with maximum degree $Δ$, Havet, Hörsch, and Rambaud~(2024) proved $\diam(\mathcal{I}(G)) \le 2Δ-1$ and conjectured that $\diam(\mathcal{I}(G)) \le Δ$. We prove the conjecture when $Δ=3$ with the help of computer calculations.

💡 Research Summary

The paper investigates two fundamental questions concerning the inversion diameter of a graph, defined as the diameter of its inversion graph 𝕀(G). The inversion graph has as vertices all possible orientations of the underlying undirected graph G, and two orientations are adjacent if one can be obtained from the other by inverting a vertex subset (i.e., reversing all arcs whose both endpoints lie inside the subset).

Result 1 – Tightness of the tree‑width bound.
Earlier work by Havet, Hörsch, and Rambaud (2024) proved that for any graph of treewidth k, the inversion diameter satisfies diam(𝕀(G)) ≤ 2k, and they exhibited graphs with diameter k + 2, leaving a gap. The authors construct, for every positive integer k, an infinite family of graphs G(k)ₘ of treewidth k whose inversion diameter reaches the upper bound 2k, thereby showing the bound is optimal.

The construction starts with a k‑clique G(k)₀ equipped with an arbitrary edge‑labeling π(k)₀ (labels are vectors in 𝔽₂ᵏ). Recursively, for each existing k‑clique and each vector x ∈ 𝔽₂ᵏ, a new vertex u is added and connected to the clique vertices v₁,…,v_k with edge labels x₁,…,x_k respectively. This yields 2ᵏ new vertices per k‑clique, preserving the property of being a k‑tree (hence treewidth ≤ k).

A key tool is Proposition 2.1, which states that diam(𝕀(G)) ≤ t iff every labeling π admits a t‑dimensional vector assignment respecting the labeling (i.e., each vertex gets a vector in 𝔽₂ᵗ such that the dot product of the vectors of the endpoints equals the edge label). Assuming the limit λ(k)=limₘ diam(𝕀(G(k)ₘ)) were ≤ 2k − 1 leads to the existence of a (2k − 1)‑dimensional assignment for all m. The authors then develop a series of lemmas about linear independence of vectors assigned to k‑cliques (Lemmas 3.1–3.4) and introduce the notion of a “bad” p‑clique (Definition 3.5) where the intersection of its span with its orthogonal complement has large dimension. Lemma 3.6 shows that if a bad p‑clique with p < k exists, one can construct a larger bad clique, eventually contradicting the assumed dimensional bound. Consequently λ(k) cannot be ≤ 2k − 1, forcing λ(k)=2k. This confirms that the 2k bound is tight for treewidth k.

Result 2 – Confirmation of the Δ‑conjecture for Δ = 3.
Havet et al. also proved a general bound diam(𝕀(G)) ≤ 2Δ − 1 for graphs of maximum degree Δ, and conjectured the stronger diam(𝕀(G)) ≤ Δ. The conjecture is known for Δ ≤ 2. The present paper settles the case Δ = 3. Using exhaustive computer enumeration, the authors generate all possible edge labelings for graphs with maximum degree three (including subcubic graphs) and test whether a 3‑dimensional vector assignment exists for each labeling. Their program verifies that such an assignment always exists, which by Proposition 2.1 yields diam(𝕀(G)) ≤ 3. Thus the conjectured bound holds for Δ = 3. The authors note that a purely combinatorial proof remains an open challenge.

Methodological contributions.
The work showcases a powerful algebraic framework that translates the combinatorial problem of inversion sequences into linear algebra over the binary field. By linking inversion diameter to the existence of low‑dimensional vector assignments, the authors can exploit concepts such as orthogonal complements, self‑orthogonal subspaces, and dimension counting. The “bad clique” argument provides a novel way to force contradictions when the assumed dimension is too low. Additionally, the paper demonstrates how computer‑assisted exhaustive search can settle conjectures for small degree classes, while still motivating the search for a theoretical proof.

Implications and future directions.
The tightness result for treewidth suggests that any improvement on the general bound must exploit additional graph structure beyond treewidth. The confirmation of the Δ‑conjecture for Δ = 3 encourages further investigation for higher Δ, possibly via refined algebraic techniques or more sophisticated computational methods. Moreover, the algebraic perspective may be applicable to related reorientation problems, such as the inversion number (minimum inversions to reach an acyclic orientation) or to other transformation graphs defined by local operations. Overall, the paper advances our understanding of how structural parameters constrain the complexity of orientation transformations.