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 $G_1, \dots ,G_k$，其中任意两个不同的 $G_i$ 与 $G_j$（$i \neq j$）共享同一个图 $G$，问题“固定边的同时嵌入”（SEFE）询问是否存在每个输入图的平面绘制，使得所有绘制在 $G$ 上重合。虽然对于两个输入图的情况，SEFE 仍是开放问题，但对于 $k \geq 3$，该问题是 NP-完全的 [Schaefer, JGAA 13]。本文中，我们探讨 SEFE 的参数化复杂性。我们证明，当参数为 $k$ 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时，SEFE 是固定参数可解的（FPT）。关于共享图 $G$，我们表明即使 $G$ 是最大度为 4 的树，SEFE 也是 NP-完全的。结合已知的 NP-难度归约 [Angelini et al., TCS 15]，这使我们能够推断出 $G$ 的若干参数（包括最大度、度数为 1 的邻居的最大数量、顶点覆盖数以及割点数）是难解的。我们还确定了这些参数所有配对的可解性：我们给出了顶点覆盖数与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，而所有剩余组合被证明是难解的。