Constrained Simultaneous and Near-Simultaneous Embeddings - Frati, Fabrizio; Kaufmann, Michael; Kobourov, Stephen (Journal of Graph Algorithms and Applications, 2009)
A geometric simultaneous embedding of two graphs G1=(V1,E1) and G2=(V2,E2) with a bijective mapping of their vertex sets γ:V1 → V2 is a pair of planar straight-line drawings Γ1 of G1 and Γ2 of G2, such that each vertexv2=γ(v1), with v1 ∈ V1 and v2 ∈ V2, 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 assumingthat the input graphs do not share common edges does not yield larger classes of graphs that can be simultaneously embedded. Further, if a prescribedcombinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open in the standard geometricsimultaneous embedding setting. Finally, we present some results on the near-simultaneous embedding problem, in which vertices are not forced to beplaced exactly at the same, but just at "nearby" points in different drawings. <!---->