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

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Abstract

A set of planar graphs {G1(V,E1),⋯, Gk(V,Ek)} 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 Ei does not cross any other edge in Ei (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 Gi whose edge set Ei contains the edge (u,v). We give a necessary and sufficient condition for two graphs whose union is homeomorphic to K5 or K3 ,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(n4)-time algorithms to compute a SEFE. © 2011 Elsevier B.V.

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Fowler, J. J., Jünger, M., Kobourov, S. G., & Schulz, M. (2011). Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges. Computational Geometry: Theory and Applications, 44(8), 385–398. https://doi.org/10.1016/j.comgeo.2011.02.002

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