Inversion diameter and treewidth

Abstract

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 $\mathrm{diam}(\mathcal{I}(G))$.Havet, Hörsch, and Rambaud~(2024) first proved that for $G$ of treewidth $k$, $\mathrm{diam}(\mathcal{I}(G)) \le 2k$, 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 $\mathrm{diam}(\mathcal{I}(G)) \le 2k$ is tight.Moreover, for graphs with maximum degree $Δ$, Havet, Hörsch, and Rambaud~(2024) proved $\mathrm{diam}(\mathcal{I}(G)) \le 2Δ-1$ and conjectured that $\mathrm{diam}(\mathcal{I}(G)) \le Δ$. We prove the conjecture when $Δ=3$ with the help of computer calculations.