Intersection Graphs in Simultaneous Embedding with Fixed Edges

We examine the simultaneous embedding with fixed edges problem for two planar graphs G1 and G2 with the focus on their intersection S = G1∩G2. In particular, we will present the complete set of intersection graphs S that guarantee a simultaneous embedding with fixed edges for (G1, G2). More formally, we define the subset ISEFE of all planar graphs as follows: A graph S lies in ISEFE if every pair of planar graphs (G1, G2) with intersection S = G1 ∩ G2 has a simultaneous embedding with fixed edges. We will characterize this set by a detailed study of topological embeddings and finally give a complete list of graphs in this set as our main result of this paper. Submitted: March 2009 Reviewed: June 2009 Revised: July 2009 Accepted: July 2009 Final: July 2009 Published: July 2009 Article type: Regular paper Communicated by: S. G. Kobourov Research supported by the German Science Foundation, Grant DFG(JU204/11-1) E-mail addresses: [email protected] (Michael Jünger) [email protected] (Michael Schulz) 206 M. Jünger and M. Schulz Intersection graphs in SEFE