A note concerning paths and independence number in digraphs (Q913811)

The authors show that there exist digraphs D such that for all paths \(P_ 1\) and \(P_ 2\) we \(\alpha (D\setminus (P_ 1\cup P_ 2))=\alpha (D)\) They believe that \(f(k)=k(f(k))\) is the smallest integer such that if D is any digraph with \(\alpha (D)=k\) then D contains f(k) paths \(P_ i\) such that \(\alpha (D\setminus \cup_{i=1}^{f(k)}P_ i)<\alpha (D))\).