Constrained simultaneous and near-simultaneous embeddings

A geometric simultaneous embedding of two graphs G ₁=(V ₁,E ₁) and G ₂=(V ₂,E ₂) with a bijective mapping of their vertex sets γ:V ₁ → V ₂ is a pair of planar straight-line drawings Γ ₁ of G ₁ and Γ ₂ of G ₂, such that each vertex v2=γ(v1), with v1 ∈ V ₁ and v2 ∈ V ₂, is mapped in Γ ₂ to the same point where v1 is mapped in Γ ₁. 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.