Inversion Diameter and Treewidth

TL;DR

This work investigates the inversion diameter of graphs, defined via inversions on vertex subsets and the resulting inversion graph $\mathcal{I}(G)$. It shows the upper bound $diam(\mathcal{I}(G)) \le 2k$ for graphs of treewidth $k$ is tight by constructing a family $G_m^{(k)}$ with $tw(G_m^{(k)})\le k$ and $diam(\mathcal{I}(G_m^{(k)}))=2k$ for large $m$, thereby answering a question of Havet et al. It also resolves the subcubic case of a conjecture by computer-assisted proofs, establishing $diam(\mathcal{I}(G)) \le 3$ for all graphs with maximum degree $\Delta=3$. Together, the results delineate the inversion-diameter landscape across treewidth and degree, providing tight bounds and a complete verification for the cubic case.

Abstract

In an oriented graph $\overrightarrow{G}$, the inversion of a subset $X$ of vertices consists in reversing 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 $X$ 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 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.

Figures (7)

• Figure 1: An example of outer-planar graph with labelled edges of inversion diameter $4$ verified by computer.
• Figure 2: $K_4^-$ in $G$.
• Figure 3: Triangle in $G$.
• Figure 4: $P_3$ with edges labelled one in $G$.
• Figure 5: $K_{2,3}$ in $G$.

Theorems & Definitions (27)

• Theorem 1.1: Havet et al. havet2024diameter
• Theorem 1.2: Havet et al. havet2024diameter
• Conjecture 1.3: Havet et al. havet2024diameter
• Theorem 1.4
• Theorem 1.5
• Proposition 2.1: havet2024diameter
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.1