Constrained Simultaneous and Near-Simultaneous Embeddings
Vol. 13, no. 3, pp. 447-465, 2009. Regular paper.
Abstract A geometric simultaneous embedding of two graphs G1=(V1,E1) and G2=(V2,E2) with a bijective mapping of their vertex sets γ:V1V2 is a pair of planar straight-line drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2=γ(v1), with v1V1 and v2V2, is mapped in Γ2 to the same point where v1 is mapped in Γ1.
In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that assuming that the input graphs do not share common edges does not yield larger classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open in the standard geometric simultaneous embedding setting. Finally, we present some results on the near-simultaneous embedding problem, in which vertices are not forced to be placed exactly at the same, but just at "nearby" points in different drawings.
Submitted: December 2007.
Reviewed: May 2008.
Revised: August 2008.
Accepted: November 2008.
Final: December 2008.
Published: November 2009.
Communicated by Seok-Hee Hong and Takao Nishizeki
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