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

  • Authors:

  • J. Joseph Fowler;Michael Jünger;Stephen G. Kobourov;Michael Schulz

  • Affiliations:

  • Department of Computer Science, University of Arizona, USA;Institut für Informatik, Universität zu Köln, Köln, Germany;Department of Computer Science, University of Arizona, USA;Institut für Informatik, Universität zu Köln, Köln, Germany

  • Venue:
  • Computational Geometry: Theory and Applications
  • Year:
  • 2011

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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.