On some-complete SEFE problems

Abstract

We investigate the complexity of some problems related to the Simultaneous Embedding with Fixed Edges (SEFE) problem which, given k planar graphs G 1,.,G k on the same set of vertices, asks whether they can be simultaneously embedded so that the embedding of each graph be planar and common edges be drawn the same. While the computational complexity of SEFE with k = 2 is a central open question in Graph Drawing, the problem is-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-complete even when (i) the intersection graph is connected and (ii) two of the three input graphs are biconnected. This result implies that the Partitioned T-Coherent k-Page Book-Embedding is-complete with k ≥ 3, which was only known for k unbounded [Hoske, Bachelor Thesis (2012)]. Further, we prove that the problem of maximizing the number of edges that are drawn the same in a SEFE of two graphs is-complete (optimization of SEFE, Open Problem 9, Chapter 11 of the Handbook of Graph Drawing and Visualization). © 2014 Springer International Publishing Switzerland.