On the $(\leq p)$-inversion diameter of oriented graphs

2026-04-06

AI Summary

This study investigates the $(\leq p)$-reversal diameter of directed graphs, defined as the maximum number of steps required to transform one orientation into another using at most $p$ vertex reversals per step. By constructing the $(\leq p)$-reversal graph and analyzing its structure through techniques from graph orientation, combinatorial analysis, and structural decomposition, the authors establish the first general upper bound for arbitrary graphs, expressed in terms of a parameter $\Psi_p$. For trees, they derive tight exact results, notably showing that $id^{\leq3}_{\mathcal{F}}(n)=\lceil(n-1)/2\rceil$. Furthermore, for planar graphs, they prove a polynomial upper bound dependent on $p$, substantially improving upon existing asymptotic estimates.

Abstract

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 $p\geq 6$.

Problem

• oriented graphs
• inversion diameter
• graph orientation
• diameter
• combinatorial reconfiguration

Innovation

• inversion diameter
• oriented graphs
• graph reorientation
• planar graphs
• combinatorial bounds