SEFE without Mapping via Large Induced Outerplane Graphs in Plane Graphs

We show that every n-vertex planar graph admits a simultaneous embedding without mapping and with fixed edges with any (n/2)vertex planar graph. In order to achieve this result, we prove that every n-vertex plane graph has an induced outerplane subgraph containing at least n/2 vertices. Also, we show that every n-vertex planar graph and every n-vertex planar partial 3-tree admit a simultaneous embedding without mapping and with fixed edges.