🤖 AI Summary
This paper investigates the parameterized complexity of Simultaneous Embedding with Fixed Edges (SEFE): given $k$ planar input graphs that share a common subgraph $G$, decide whether they admit planar embeddings that agree on $G$. Addressing this long-standing open problem, we establish the first fixed-parameter tractability (FPT) results for SEFE with respect to the combined parameters $k + mathrm{vc}(G^cup)$ and $k + mathrm{fe}(G^cup)$, where $mathrm{vc}$ and $mathrm{fe}$ denote vertex cover number and feedback edge set size of the union graph $G^cup$, respectively. We further prove NP-completeness even when $G$ is a tree of maximum degree at most four. Systematically analyzing six structural graph parameters of $G$—including vertex cover number, cut-vertex number, and maximum degree—we fully classify the computational complexity of SEFE under all 15 possible dual-parameter combinations. Our techniques integrate parameterized algorithm design, intricate reduction constructions, structural graph analysis, and dynamic programming over vertex covers and feedback edge sets.
📝 Abstract
Given $k$ input graphs $G_1, dots ,G_k$, where each pair $G_i$, $G_j$ with $i
eq j$ 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.
