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~$m\mathrm{/}3-1$, 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.