A linear algorithm for embedding planar graphs using PQ-trees
Journal of Computer and System Sciences
An Optimal Algorithm for Euclidean Shortest Paths in the Plane
SIAM Journal on Computing
Embedding Planar Graphs at Fixed Vertex Locations
GD '98 Proceedings of the 6th International Symposium on Graph Drawing
On simultaneous planar graph embeddings
Computational Geometry: Theory and Applications
Embedding graphs simultaneously with fixed edges
GD'06 Proceedings of the 14th international conference on Graph drawing
Simultaneous geometric graph embeddings
GD'07 Proceedings of the 15th international conference on Graph drawing
Simultaneous graph embeddings with fixed edges
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Two trees which are self–intersecting when drawn simultaneously
GD'05 Proceedings of the 13th international conference on Graph Drawing
Colored simultaneous geometric embeddings
COCOON'07 Proceedings of the 13th annual international conference on Computing and Combinatorics
Testing planarity of partially embedded graphs
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
On a tree and a path with no geometric simultaneous embedding
GD'10 Proceedings of the 18th international conference on Graph drawing
IWOCA'10 Proceedings of the 21st international conference on Combinatorial algorithms
Computational Geometry: Theory and Applications
Journal of Discrete Algorithms
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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 whether a SEFE exists for pairs of graphs whose union is homeomorphic to K 5 or K 3,3 . 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 efficient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE.