On Simultaneous Graph Embedding

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another where the mapping is not given. In particular, we show that without mapping, any number of outerplanar graphs can be embedded simultaneously on an $O(n)\times O(n)$ grid, and an outerplanar and general planar graph can be embedded simultaneously on an $O(n^2)\times O(n^3)$ grid. If the mapping is given, we show how to embed two paths on an $n \times n$ grid, a caterpillar and a path on an $n \times 2n$ grid, or two caterpillar graphs on an $O(n^2)\times O(n^3)$ grid. We also show that 5 paths, or 3 caterpillars, or two general planar graphs cannot be simultaneously embedded given the mapping.