Which graph problems are $W[1]$-Hard on directed(/weighted) graphs but FPT on undirected(/unweighted) graphs? Following the equivalent questions regarding NP-Completeness (see the weight question and the directed question), I was wondering how parameterized problems are affected by these attributes.


  
  Which $NP$-hard graph problems are $W[1]$-Hard on directed graphs, but fixed parameter tractable on undirected graphs?
  





  
  Which $NP$-hard graph problems are $W[1]$-Hard on weighted graphs, but fixed parameter tractable on unweighted graphs?
  




OK, so we have problems which becomes harder on the directed version. What about weights? Can they make parameterized problem harder?
The disjoint paths problem: given $G$ and $k$ pairs of nodes, are there node disjoint paths connecting the given pairs. Parameterized by $k$, in FPT when $G$ is undirected from the seminal work of Robertson and Seymour. NP-Hard for $k=2$ when $G$ is directed - from work of Fortune, Hopcroft and Wylie (1980).