Abstract. 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 I SEFE of all planar graphs as follows: A graph S lies in I SEFE 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.