The inversion number of dijoins and blow-up digraphs

TL;DR

This paper investigates the inversion number of oriented graphs, disproves existing conjectures, and constructs specific tournaments with minimal inversion numbers using blow-up graphs.

Contribution

It proves a new relation for the inversion number when adding a directed triangle and disprove existing conjectures, also constructing tournaments with minimal inversion numbers.

Findings

Disproved conjectures by Aubian et al. and Alon et al.

Established that adding a directed triangle increases the inversion number by one.

Constructed tournaments with inversion number approximately n/3 using blow-up graphs.

Abstract

For an oriented graph $D$, the $inversion$ of $X \subseteq V(D)$ in $D$ is the digraph obtained from $D$ by reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by $inv(D)$, is the minimum number of inversions needed to transform $D$ into an acyclic digraph. In this paper, we first show that $inv (\overrightarrow{C_3} \Rightarrow D)= inv(D) +1$ for any oriented graph $\textit{D}$ with even inversion number $inv(D)$, where the dijoin $\overrightarrow{C_3} \Rightarrow D$ is the oriented graph obtained from the disjoint union of $\overrightarrow{C_3}$ and $D$ by adding all arcs from $\overrightarrow{C_3}$ to $D$. Thus we disprove the conjecture of Aubian el at. \cite{2212.09188} and the conjecture of Alon el at. \cite{2212.11969}. We also study the blow-up graph which is an oriented graph obtained from a tournament by replacing all vertices into oriented graphs. We construct a tournament $T$ with order $n$ and $inv(T)=\frac{n}{3}+1$ using blow-up graphs.