  Upward planarity testing and Rectilinear planarity testing are central
problems in graph drawing. It is known that they are both NP-complete, but XP
when parameterized by treewidth. In this paper we show that these two problems
are W[1]-hard parameterized by treewidth, which answers open problems posed in
two earlier papers. The key step in our proof is an analysis of the
All-or-Nothing Flow problem, a generalization of which was used as an
intermediate step in the NP-completeness proof for both planarity testing
problems. We prove that the flow problem is W[1]-hard parameterized by
treewidth on planar graphs, and that the existing chain of reductions to the
planarity testing problems can be adapted without blowing up the treewidth. Our
reductions also show that the known $n^{O(tw)}$-time algorithms cannot be
improved to run in time $n^{o(tw)}$ unless ETH fails.
