Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG−X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.
Mathematics Subject Classification (1991)05 C 75 05 C 70 90 C 10 90 C 27
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