Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for when a pair of graphs whose union is homeomorphic to K5 or K3,3 can have an SEFE. This allows us to determine which (outer)planar graphs always an SEFE with any other (outer)planar graphs. In both cases, we provide efficient algorithms to compute the simultaneous drawings. Finally, we provide an linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has an SEFE.

• Type: Journal
• Year: 2011
• Where: COMGEO
• Authors: J. Joseph Fowler, Michael Jünger, Stephen G. Kobourov, Michael Schulz
• Keywords: Applied Computing, COMGEO 2011, Linear-time Decision Algorithm, Planar Graphs, Simultaneous Drawings