  The problem Level Planarity asks for a crossing-free drawing of a graph in
the plane such that vertices are placed at prescribed y-coordinates (called
levels) and such that every edge is realized as a y-monotone curve. In the
variant Constrained Level Planarity, each level y is equipped with a partial
order <_y on its vertices and in the desired drawing the left-to-right order of
vertices on level y has to be a linear extension of <_y. Constrained Level
Planarity is known to be a remarkably difficult problem: previous results by
Klemz and Rote [ACM Trans. Alg. 2019] and by Br\"uckner and Rutter [SODA 2017]
imply that it remains NP-hard even when restricted to graphs whose tree-depth
and feedback vertex set number are bounded by a constant and even when the
instances are additionally required to be either proper, meaning that each edge
spans two consecutive levels, or ordered, meaning that all given partial orders
are total orders. In particular, these results rule out the existence of
FPT-time (even XP-time) algorithms with respect to these and related graph
parameters (unless P=NP). However, the parameterized complexity of Constrained
Level Planarity with respect to the vertex cover number of the input graph
remained open.
  In this paper, we show that Constrained Level Planarity can be solved in
FPT-time when parameterized by the vertex cover number. In view of the previous
intractability statements, our result is best-possible in several regards: a
speed-up to polynomial time or a generalization to the aforementioned smaller
graph parameters is not possible, even if restricting to proper or ordered
instances.
