Abstract
Given k input graphs 
, where each pair 
with \(i \ne j\) 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 \(k \ge 3\) [18].
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 union graph 
. 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 [1], 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.
Funded by the Deutsche Forschungsgemeinschaft (German Research Foundation, DFG) under grant RU-1903/3-1.
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Fink, S.D., Pfretzschner, M., Rutter, I. (2023). Parameterized Complexity of Simultaneous Planarity. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14466. Springer, Cham. https://doi.org/10.1007/978-3-031-49275-4_6
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