Minimum Feedback Arc and Vertex Sets of a Directed Graph

To determine a minimum set of arcs of an arbitrary directed graph which, if removed, leave the graph without directed circuits, is an outstanding problem in graph theory. A related problem is that of finding a minimum set of vertices which, if removed together with their incident arcs, leave the graph with no directed circuits. A closed form solution of both problems is presented. The determination of those minimum sets for a graph with n vertices involves the expansion of an n-th order permanent and some algebraic manipulations of the resultant expression, subject to the absorption laws of Boolean algebra. The proposed procedure renders all possible solutions simultaneously.