In this seminar, we will present some results regarding the problem of inversions on oriented graphs. Let D be an oriented graph and let X be a subset of vertices of D. The inversion of X consists in the operation of reversing the orientation of every arc of D with both extremes in X. An inversion of size exactly p (respec. at most p) is called a (=p)-inversion (respec. (<=p)-inversion). An oriented graph is (=p)-invertible (respec. (<=p)-invertible) if it can be made acyclic using (=p)-inversions (respec. (<=p)-inversions). We first deal with the problem of dec,iding whether an oriented graph is (=p)-invertible. We will present a brief outline of the results showing that this problem is NP-complete when p=n-1 and polynomial-time solvable otherwise. We then consider the
(=p)-inversion number (respec. (<=p)-inversion number) of an oriented graph D, which is the minimum number of (=p)-inversions (respec. (<=p)-inversions) rendering D acyclic. We will present some upper bounds for both parameters in terms of the size of a minimum feedback arc set of D. Finally, we will present some results regarding the complexity of deciding whether the (=p)-inversion number or (<=p)-inversion number is at most k. This is a joint work with Jørgen Bang-Jensen, Frédéric Havet, Florian Hörsch, Clément Rambaud, and Amadeus Reinald.