Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic.
@cite_3 consider the case where the embedding of each of the input graphs is already fixed. With this restriction SEFE becomes trivial for two graphs since it remains to test whether the two graphs induce the same embedding on the common graph. They show that it can also be decided efficiently for three graphs. However, it becomes NP-hard for at least fourteen graphs. They also consider the problem SGE for the case that the embedding of each graph is fixed and show that it is NP-hard for at least thirteen graphs.
@cite_3
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Given k planar graphs G1,…,Gk, deciding whether they admit a simultaneous embedding with fixed edges (Sefe ) and whether they admit a simultaneous geometric embedding (Sge ) are NP-hard problems, for k≥3 and for k≥2, respectively. In this paper we consider the complexity of Sefe and of Sge when the graphs G1,…,Gk have a fixed planar embedding. In sharp contrast with the NP-hardness of Sefe for three non-embedded graphs, we show that Sefe is polynomial-time solvable for three graphs with a fixed planar embedding. Furthermore, we show that, given k embedded planar graphs G1,…,Gk, deciding whether a Sefe of G1,…,Gk exists and deciding whether an Sge of G1,…,Gk exists are NP-hard problems, for k≥14 and k≥13, respectively.