In this work we investigate the complexity of some combinatorial problems related to the Simultaneous Embedding with Fixed Edges (SEFE) and the Partitioned T-Coherent k-Page Book Embedding (PTBE-k) problems, which are known to be equivalent under certain conditions. Given k planar graphs on the same set of n vertices, the SEFE problem asks to find a drawing of each graph on the same set of n points in such a way that each edge that is common to more than one graph is represented by the same curve in the drawings of all such graphs. Given a tree T with n leaves and a collection of k edge-sets E<inf>i</inf> connecting pairs of leaves of T, the PTBE-k problem asks to find an ordering O of the leaves of T that is represented by T such that the endvertices of two edges in any set E<inf>i</inf> do not alternate in O.The SEFE problem is NP-complete for k≥3 even if the intersection graph is the same for each pair of graphs (sunflower intersection). We prove that this is true even when the intersection graph is a tree and all the input graphs are biconnected. This result implies the NP-completeness of PTBE-k for k≥3. However, we prove stronger results on this problem, namely that PTBE-k remains NP-complete for k≥3 even if (i) two of the input graphs G<inf>i</inf>=T∪E<inf>i</inf> are biconnected and T is a caterpillar or if (ii) T is a star. This latter setting is also known in the literature as Partitioned k-Page Book Embedding. On the positive side, we provide a linear-time algorithm for PTBE-k when all but one of the edge-sets induce connected graphs.Finally, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs (. optimization of SEFE) is NP-complete, even in several restricted settings.
Angelini, P., Da Lozzo, G., & Neuwirth, D. (2015). Advancements on SEFE and Partitioned Book Embedding problems. Theoretical Computer Science, 575(1), 71–89. https://doi.org/10.1016/j.tcs.2014.11.016