A note on the invertibility of oriented graphs

For an oriented graph D and a set X⊆ V(D), the inversion of X in D is the graph obtained from D by reversing the orientation of each edge that has both endpoints in X. Define the inversion number of D, denoted $inv(D)$, to be the minimum number of inversions required to obtain an acyclic oriented graph from D. We show that inv(D₁ → D₂) > inv(D₁), for any oriented graphs D₁ and D₂ such that inv(D₁) = inv(D₂) ≥ 1. This resolves a question of Aubian, Havet, H\"{o}rsch, Klingelhoefer, Nisse, Rambaud and Vermande. Our proof proceeds via a natural connection between the graph inversion number and the subgraph complementation number.