Diameter of the inversion graph

TL;DR

This work introduces the labelled inversion graph $\mathcal{I}(G)$ and its diameter $diam(\mathcal{I}(G))$, establishing a deep link between inversion diameter and key colouring parameters: star chromatic number $\chi_s$, acyclic chromatic number $\chi_a$, and oriented chromatic number $\chi_o$. It develops an algebraic (GF$(2)$) framework to bound or realize $diam(\mathcal{I}(G))$, and derives sharp bounds for a variety of graph classes, including planar graphs ($diam\le 12$), planar girth constraints, graphs with bounded maximum degree ($diam\le 2\Delta-1$), and graphs with bounded treewidth ($diam\le 2t$). The paper also shows that computing the inversion diameter is NP-hard and provides improved, explicit upper bounds in terms of $\chi_s$, $\chi_a$, and $\chi_o$, as well as in terms of maximum degree and treewidth. Together, these results identify a rich interplay between inversion operations and classical graph colourings, and they map out several open questions for minor-closed and planar graph families.

Abstract

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 $Δ$ ($diam(\mathcal{I}(G)) \leq 2Δ-1$), 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.

Figures (5)

• Figure 1: Some cases in the proof of Lemma \ref{['lem:good-ordering']}. The vertices are placed from left to right according to $\sigma$. The edges in $E(G') \setminus E(G)$ and the vertices in $V(G')$ are depicted in red.
• Figure 2: The case $i<j,k<l,j<l$ in the proof of Lemma \ref{['lem:good-ordering']}, with vertices placed from left to right according to $\sigma$. Since the relative position of $w_1$ with $t_1$ and $t_2$ is unknown, $w_1$ is placed below. The edges in $E(G') \setminus E(G)$ and the vertices in $V(G')$ are depicted in red.
• Figure 3: Construction of a graph with treewidth $2$ and inversion diameter $4$.
• Figure 4: A planar graph $G$ with inversion diameter at least $5$. More precisely, if $\pi \colon E(G) \to \mathbb{F}_2$ is the labelling function depicted, then there is no family $(\mathbf{u})_{u \in V(G)} \in \mathbb{F}_2^4$ such that $\pi(uv) = \mathbf{u} \cdot \mathbf{v}$ for every $uv \in E(G)$.
• Figure 5: A $5$-regular graph $G$ with inversion diameter $4$. Since this graph contains $K_5$, Theorem \ref{['thm:multipartite']} implies $\mathop{\mathrm{diam}}\nolimits(\mathcal{I}(G)) \geqslant 4$. The other inequality $\mathop{\mathrm{diam}}\nolimits(\mathcal{I}(G)) \leqslant 4$ was checked by computer.

Theorems & Definitions (103)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• proof
• Theorem 2.3
• proof
• Proposition 3.1
• proof
• Theorem 3.2