Research has concentrated on finding drawings with few bends, or with few crossings between edges from the two graphs.
In particular, the thickness of a given graph is two, if the graph's edges can be partitioned into two subgraphs that have a simultaneous embedding, and the geometric thickness is two, if the edges can be partitioned into two subgraphs with simultaneous geometric embedding.
One of the most basic results of this type is the fact that any two path graphs on the same vertex set always have a simultaneous embedding.
This type of drawing places the vertices in an integer lattice of dimensions linear in the graph sizes.
However, pairs of planar graphs and a matching, or of a Angelini, Geyer, Neuwirth and Kaufmann showed that a tree and a path exist, that have no simultaneous geometric embedding.
The proof of this result also implies that for some pairs of graphs that have simultaneous geometric embeddings, the smallest grid on which they can be drawn has doubly exponential size.
It is also possible to find a simultaneous embedding with fixed edges for any pair of a planar graph and a tree.
[7][8][9] It is an open question whether the existence of a simultaneous embedding with fixed edges for two given graphs can be tested in polynomial time.