In an oriented graph $\vec{G}$, the inversion of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endvertices in $X$. The inversion graph of a labelled graph $G$, denoted by ${\mathcal{I}}(G)$, is the graph whose vertices are the labelled orientations of $G$ in which two labelled orientations $\vec{G}_1$ and $\vec{G}_2$ of $G$ are adjacent if and only if there is an inversion $X$ transforming $\vec{G}_1$ into $\vec{G}_2$. In this paper, we study the inversion diameter of a graph which is the diameter of its inversion graph denoted by $diam(\mathcal{I}(G))$. We show that the inversion diameter is tied to the star chromatic number, the acyclic chromatic number and the oriented chromatic number. Thus a graph class has bounded inversion diameter if and only if it also has bounded star chromatic number, acyclic chromatic number and oriented chromatic number. We give some upper bounds on the inversion diameter of a graph $G$ contained in one of the following graph classes: planar graphs ($diam(\mathcal{I}(G)) \leq 12$), planar graphs of girth 8 ($diam(\mathcal{I}(G)) \leq 3$), graphs with maximum degree $\Delta$ ($diam(\mathcal{I}(G)) \leq 2\Delta -1$), graphs with treewidth at mots $t$ ($diam(\mathcal{I}(G)) \leq 2t$). We also show that determining the inversion diameter of a given graph is NP-hard.

翻译：在定向图 $\vec{G}$ 中，顶点子集 $X$ 的反转操作是指反转所有两端点均属于 $X$ 的弧的方向。对于标号图 $G$，其反转图 ${\mathcal{I}}(G)$ 定义为顶点集为 $G$ 的所有标号定向的图，其中两个标号定向 $\vec{G}_1$ 与 $\vec{G}_2$ 相邻当且仅当存在一个反转 $X$ 可将 $\vec{G}_1$ 变换为 $\vec{G}_2$。本文研究图的"反转直径"，即其反转图的直径，记为 $diam(\mathcal{I}(G))$。我们证明了反转直径与星色数、无环色数以及定向色数密切相关。因此，图类的反转直径有界当且仅当其星色数、无环色数和定向色数均有界。我们给出了以下图类中图 $G$ 的反转直径的上界：平面图（$diam(\mathcal{I}(G)) \leq 12$）、围长至少为8的平面图（$diam(\mathcal{I}(G)) \leq 3$）、最大度数为 $\Delta$ 的图（$diam(\mathcal{I}(G)) \leq 2\Delta -1$）、树宽不超过 $t$ 的图（$diam(\mathcal{I}(G)) \leq 2t$）。此外，我们还证明了确定给定图的反转直径是NP困难的。