Advancements on SEFE and Partitioned Book Embedding problems

In this work we investigate the complexity of some problems related to the 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 ≥ 3 [Gassner et al., WG ’06], even if the intersection graph is the same for each pair of graphs (sunflower intersection) [Schaefer, JGAA (2013)]. We improve on these results by proving that SEFE with k ≥ 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 ≥ 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 PTBEk 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 isNP-complete in several restricted settings (optimization version of SEFE, Open Problem 9, Chapter 11 of the Handbook of Graph Drawing and Visualization).