Inversion Diameter of Planar Graphs

Abstract

Given an oriented graph $\vec{G}$ and a subset of vertices $X\subseteq V(\vec{G})$, the \emph{inversion} of $X$ is the operation that reverses the orientation of every arc with both endpoints in $X$. For a simple graph $G$, the inversion diameter $\mathrm{diam}(I(G))$ is the maximum distance between two orientations of $G$ under inversions of vertex sets. We prove the sharp bound \[ \operatorname{diam}(I(G))\le 2χ_a(G)-2, \] where ${\chi }_a(G)$ is the acyclic chromatic number. Consequently, every planar graph has inversion diameter at most $8$, improving the previously known bound $12$. Using strong-degeneracy arguments, we also obtain upper bounds $7$, $5$, and $4$ for planar graphs of girth at least $4$, $5$, and $6$, respectively.