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.

翻译：给定$k$个输入图$G_1, \dots ,G_k$，其中每对$i \neq j$的图$G_i$和$G_j$共享同一个图$G$，固定边同时嵌入问题（SEFE）询问：是否每个输入图都存在一个平面绘制，使得所有绘制在$G$上重合。虽然对于两个输入图的情形，SEFE问题仍未解决，但对于$k \geq 3$的情况，该问题是NP完全的[Schaefer, JGAA 13]。本文探索了SEFE的参数化复杂性。我们证明，SEFE关于参数$k$加上并图$G^\cup = G_1 \cup \dots \cup G_k$的顶点覆盖数或反馈边集数是固定参数可处理的（FPT）。关于共享图$G$，我们证明即使$G$是最大度为4的树，SEFE也是NP完全的。结合已知的NP困难归约[Angelini et al., TCS 15]，可以断定$G$的多个参数（包括最大度、度数为1的邻居的最大数目、顶点覆盖数和割点数）是难解的。我们还确定了这些参数所有对的易解性。我们给出了顶点覆盖数加上前两个参数中任意一个的FPT算法，以及割点数加上最大度的FPT算法，同时证明其余所有组合都是难解的。