Diameter of the inversion graph

TL;DR

This paper investigates the inversion diameter of graphs, linking it to various chromatic numbers, providing bounds for specific graph classes, and proving that computing it is NP-hard.

Contribution

It establishes a connection between inversion diameter and chromatic numbers, provides bounds for various graph classes, and proves the NP-hardness of computing the inversion diameter.

Findings

Inversion diameter is related to star, acyclic, and oriented chromatic numbers.

Bounded inversion diameter characterizes certain graph classes.

Computing the inversion diameter is NP-hard.

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