Bounding the feedback vertex number of digraphs in terms of vertex degrees

The Turán bound (Turán (1941) ) is a famous result in graph theory, which relates the independence number of an undirected graph to its edge density. Also the Caro–Wei inequality (Caro (1979) and Wei (1981) ), which gives a more refined bound in terms of the vertex degree sequence of a graph, might be regarded today as a classical result. We show how these statements can be generalized to directed graphs, thus yielding a bound on directed feedback vertex number in terms of vertex out-degrees and in terms of average out-degree, respectively.