In an oriented graph $\vec{G}$, the {\it inversion} of a subset $X$ of vertices consists in reversing the orientation of all arcs with both endvertices in $X$. The {\it $(\leq p)$-inversion graph} of a labelled graph $G$, denoted by ${\mathcal{I}}^{\leq p}(G)$, 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 a set $X$ with $|X|\leq p$ whose inversion transforms $\vec{G}_1$ into $\vec{G}_2$. In this paper, we study the {\it $(\leq p)$-inversion diameter} of a graph, denoted by $\mathrm{id}^{\leq p}(G)$, which is the diameter of its $(\leq p)$-inversion graph. We show that there exists a smallest number $Ψ_p$ with $\frac{1}{4}p - \frac{3}{2} \leq Ψ_p \leq \frac{1}{2}p^2$ such that $\mathrm{id}^{\leq p}(G) \leq \left\lceil\frac{|E(G)|}{\lfloor p/2\rfloor}\right \rceil + Ψ_p$ for all graph $G$. We then establish better upper bounds for several families of graphs and in particular trees and planar graphs. Let us denote by $\mathrm{id}^{\leq p}_{\cal F}(n)$ (resp. $\mathrm{id}^{\leq p}_{\cal P}(n)$) the maximum $(\leq p)$-inversion diameter of a tree (resp. planar graph) of order $n$. For trees, we show $\mathrm{id}^{\leq 3}_{\cal F}(n) = \left\lceil \frac{n-1}{2}\right\rceil$, $\mathrm{id}^{\leq 4}_{\cal F}(n)=\frac{3}{8}n + Θ(1)$, $\mathrm{id}^{\leq 5}_{\cal F}(n)= \frac{2}{7}n + Θ(1)$, and $\mathrm{id}^{\leq p}_{\cal F}(n) \leq \frac{n-1}{p- c\sqrt{p}} + 2$ with $c = \sqrt{2 + \sqrt{2}}$ for all $p\geq 6$. For planar graphs, we prove $\mathrm{id}^{\leq 3}_{\cal P}(n) \leq \frac{11n}{6} - \frac{8}{3}$, $\mathrm{id}^{\leq 4}_{\cal P}(n) \leq \frac{4n}{3} + \frac{10}{3}$, and $\mathrm{id}^{\leq p}_{\cal P}(n) \leq \left\lceil\frac{3n-6}{\lfloor p/2\rfloor}\right \rceil + 8\lfloor p/2\rfloor - 8$ for all $p\geq 6$.

翻译：在有向图$\vec{G}$中，对顶点子集$X$进行{\it 翻转}是指反转所有两端点均在$X$中的弧的方向。对于一个标记图$G$，其{\it $(\leq p)$-翻转图}记作${\mathcal{I}}^{\leq p}(G)$，该图的顶点是$G$的所有标记定向，其中两个标记定向$\vec{G}_1$和$\vec{G}_2$相邻当且仅当存在一个大小$|X|\leq p$的集合$X$，使得对$X$的翻转将$\vec{G}_1$变换为$\vec{G}_2$。本文研究图的{\it $(\leq p)$-翻转直径}，记作$\mathrm{id}^{\leq p}(G)$，即其$(\leq p)$-翻转图的直径。我们证明存在最小数$Ψ_p$满足$\frac{1}{4}p - \frac{3}{2} \leq Ψ_p \leq \frac{1}{2}p^2$，使得对所有图$G$有$\mathrm{id}^{\leq p}(G) \leq \left\lceil\frac{|E(G)|}{\lfloor p/2\rfloor}\right \rceil + Ψ_p$。随后我们为几类图（特别是树和平面图）建立了更优的上界。记$\mathrm{id}^{\leq p}_{\cal F}(n)$（相应地$\mathrm{id}^{\leq p}_{\cal P}(n)$）为$n$阶树（相应地$n$阶平面图）的最大$(\leq p)$-翻转直径。对于树，我们证明$\mathrm{id}^{\leq 3}_{\cal F}(n) = \left\lceil \frac{n-1}{2}\right\rceil$，$\mathrm{id}^{\leq 4}_{\cal F}(n)=\frac{3}{8}n + Θ(1)$，$\mathrm{id}^{\leq 5}_{\cal F}(n)= \frac{2}{7}n + Θ(1)$，并且对所有$p\geq 6$有$\mathrm{id}^{\leq p}_{\cal F}(n) \leq \frac{n-1}{p- c\sqrt{p}} + 2$，其中$c = \sqrt{2 + \sqrt{2}}$。对于平面图，我们证明对所有$p\geq 6$有$\mathrm{id}^{\leq 3}_{\cal P}(n) \leq \frac{11n}{6} - \frac{8}{3}$，$\mathrm{id}^{\leq 4}_{\cal P}(n) \leq \frac{4n}{3} + \frac{10}{3}$，以及$\mathrm{id}^{\leq p}_{\cal P}(n) \leq \left\lceil\frac{3n-6}{\lfloor p/2\rfloor}\right \rceil + 8\lfloor p/2\rfloor - 8$。