  Given $k$ input graphs $G_1, \dots ,G_k$, where each pair $G_i$, $G_j$ with
$i \neq j$ shares the same graph $G$, the problem Simultaneous Embedding With
Fixed Edges (SEFE) asks whether there exists a planar drawing for each input
graph such that all drawings coincide on $G$. While SEFE is still open for the
case of two input graphs, the problem is NP-complete for $k \geq 3$ [Schaefer,
JGAA 13]. In this work, we explore the parameterized complexity of SEFE. We
show that SEFE is FPT with respect to $k$ plus the vertex cover number or the
feedback edge set number of the the union graph $G^\cup = G_1 \cup \dots \cup
G_k$. Regarding the shared graph $G$, we show that SEFE is NP-complete, even if
$G$ is a tree with maximum degree 4. Together with a known NP-hardness
reduction [Angelini et al., TCS 15], this allows us to conclude that several
parameters of $G$, including the maximum degree, the maximum number of degree-1
neighbors, the vertex cover number, and the number of cutvertices are
intractable. We also settle the tractability of all pairs of these parameters.
We give FPT algorithms for the vertex cover number plus either of the first two
parameters and for the number of cutvertices plus the maximum degree, whereas
we prove all remaining combinations to be intractable.
