Towards a dual version of Woodall's conjecture for partial 3-trees

A dual version of a conjecture by Woodall asserts that, in a planar digraph, the length of a shortest dicycle equals the maximum number of pairwise disjoint feedback arc sets. We verify this conjecture for the case where the underlying graph is a 3-tree or a partial 3-tree with girth 3. Additionally, we show that every 3-tree has a feedback arc set of size at most m/3−1, where m is the number of arcs of the digraph, and this bound is tight. We further establish an upper bound on the size of a minimum feedback arc set in k -trees. Finally, we discuss some open problems and conjectures.