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 v ₂∈=∈γ(v ₁) is mapped in Γ ₂ to the same point where v ₁ is mapped in Γ ₁, where v ₁∈ ∈V ₁ and v ₂∈ ∈V ₂. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large 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 for geometric simultaneous embedding. Finally, we present some positive and negative results on the near-simultaneous embedding problem, in which vertices are not mapped exactly to the same but to "near" points in the different drawings.