The inversion number of dijoins and blow-up digraphs

For an oriented graph \(D\), the inversion of \(X ⊆ V(D)\) 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 \(\), is the minimum number of inversions needed to transform \(D\) into an acyclic digraph. In this paper, we first show that \( inv ({C_3} ⇒ D)= +1 \) for any oriented graph \(D\) with even inversion number \(\), where the dijoin \({C_3} ⇒ D\) is the oriented graph obtained from the disjoint union of \({C_3}\) and \(D\) by adding all arcs from \({C_3}\) to \(D\). Thus we disprove the conjecture of Aubian el at. {2212.09188} and the conjecture of Alon el at. {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 \(=n/3+1\) using blow-up graphs.