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Computer Science > Computational Complexity

arXiv:1311.3607 (cs)
[Submitted on 14 Nov 2013 (v1), last revised 28 Apr 2014 (this version, v2)]

Title:Advancements on SEFE and Partitioned Book Embedding Problems

Authors:Patrizio Angelini, Giordano Da Lozzo, Daniel Neuwirth
View a PDF of the paper titled Advancements on SEFE and Partitioned Book Embedding Problems, by Patrizio Angelini and Giordano Da Lozzo and Daniel Neuwirth
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Abstract:In this work we investigate the complexity of some problems related to the {\em Simultaneous Embedding with Fixed Edges} (SEFE) of $k$ planar graphs and the PARTITIONED $k$-PAGE BOOK EMBEDDING (PBE-$k$) problems, which are known to be equivalent under certain conditions.
While the computational complexity of SEFE for $k=2$ is still a central open question in Graph Drawing, the problem is NP-complete for $k \geq 3$ [Gassner {\em et al.}, WG '06], even if the intersection graph is the same for each pair of graphs ({\em sunflower intersection}) [Schaefer, JGAA (2013)].
We improve on these results by proving that SEFE with $k \geq 3$ and sunflower intersection is NP-complete even when the intersection graph is a tree and all the input graphs are biconnected. Also, we prove NP-completeness for $k \geq 3$ of problem PBE-$k$ and of problem PARTITIONED T-COHERENT $k$-PAGE BOOK EMBEDDING (PTBE-$k$) - that is the generalization of PBE-$k$ in which the ordering of the vertices on the spine is constrained by a tree $T$ - even when two input graphs are biconnected. Further, we provide a linear-time algorithm for PTBE-$k$ when $k-1$ pages are assigned a connected graph. Finally, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is NP-complete in several restricted settings ({\em optimization version of SEFE}, Open Problem $9$, Chapter $11$ of the Handbook of Graph Drawing and Visualization).
Comments: 29 pages, 10 figures, extended version of 'On Some NP-complete SEFE Problems' (Eighth International Workshop on Algorithms and Computation, 2014)
Subjects: Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
Cite as: arXiv:1311.3607 [cs.CC]
  (or arXiv:1311.3607v2 [cs.CC] for this version)
  https://doi.org/10.48550/arXiv.1311.3607
arXiv-issued DOI via DataCite

Submission history

From: Giordano Da Lozzo [view email]
[v1] Thu, 14 Nov 2013 19:00:59 UTC (671 KB)
[v2] Mon, 28 Apr 2014 13:38:21 UTC (1,048 KB)
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