Characterizing 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 of n vertices in the plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same Jordan curve in the simultaneous drawing. We give a necessary condition for when pairs of graphs can have a simultaneous embedding with fixed edges (SEFE). This allows us to determine for which (outer)planar graphs always have a SEFE with all (outer)planar graphs with O(n2lgn) time drawing algorithms. This allows us to decide in O(n) time whether a pair of biconnected outerplanar graphs has a SEFE.