Variations on the Gallai-Milgram theorem (Q1112833)

The author strengthens the Gallai-Milgram theorem for digraphs \(D\) with independence number two: ``Let \(x\) be a vertex in \(D\) such that for any vertex \(y\neq x\), there exists a path in \(D\) from \(x\) to \(y\); then \(D\) is spanned by two disjoint paths such that \(x\) is the initial vertex of one of them.'' He proposes various problems and conjectures strengthening the Gallai-Milgram theorem and the Gallai-Roy theorem for strong digraphs.