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

Abstract

A set of planar graphs { G 1 ( V , E 1 ) , … , G k ( V , E k ) } admits a simultaneous embedding if they can be drawn on the same pointset P of order n in the Euclidean plane such that each point in P corresponds one-to-one to a vertex in V and each edge in E i does not cross any other edge in E i (except at endpoints) for i ∈ { 1 , … , k } . A fixed edge is an edge ( u , v ) that is drawn using the same simple curve for each graph G i whose edge set E i contains the edge ( u , v ) . We give a necessary and sufficient condition for two graphs whose union is homeomorphic to K 5 or K 3 , 3 to admit a simultaneous embedding with fixed edges ( SEFE). This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide O ( n 4 ) -time algorithms to compute a SEFE.