Fast algorithms for finding nearest common ancestors
SIAM Journal on Computing
SIAM Journal on Computing
A Linear Time Implementation of SPQR-Trees
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
On simultaneous planar graph embeddings
Computational Geometry: Theory and Applications
Graph-Theoretic Concepts in Computer Science
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
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
Simultaneous graph embeddings with fixed edges
WG'06 Proceedings of the 32nd international conference on Graph-Theoretic Concepts in Computer Science
Disconnectivity and relative positions in simultaneous embeddings
GD'12 Proceedings of the 20th international conference on Graph Drawing
Topological morphing of planar graphs
Theoretical Computer Science
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In this paper we study the time complexity of the problem Simultaneous Embedding with Fixed Edges (Sefe), that takes two planar graphs G"1=(V,E"1) and G"2=(V,E"2) as input and asks whether a planar drawing @C"1 of G"1 and a planar drawing @C"2 of G"2 exist such that: (i) each vertex v@?V is mapped to the same point in @C"1 and in @C"2; (ii) every edge e@?E"1@?E"2 is mapped to the same Jordan curve in @C"1 and @C"2. First, we give a linear-time algorithm for Sefe when the intersection graph of G"1 and G"2, that is the planar graph G"1"@?"2=(V,E"1@?E"2), is biconnected. Second, we show that Sefe, when G"1"@?"2 is connected, is equivalent to a suitably-defined book embedding problem. Based on this equivalence and on recent results by Hong and Nagamochi, we show a linear-time algorithm for the Sefe problem when G"1"@?"2 is a star.