The inversion number of dijoins and blow-up digraphs

For an oriented graph \(D\), the \emph{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 \emph{inversion number} of \(D\), denoted by \(\invd\), is the minimum number of inversions needed to transform \(D\) into an acyclic digraph. In this paper, we first show that \( \mathrm{inv} (\overrightarrow{C_3} \Rightarrow D)= \invd +1 \) for any oriented graph \(D\) with even inversion number \(\invd\), 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 \(\invt=\frac{n}{3}+1\) using blow-up graphs.