A pair of trees without a simultaneous geometric embedding in the plane

Abstract

Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each individual graph do not cross. We consider simultaneous embeddings of two labeled trees, with predescribed vertex correspondences, and present an instance of such a pair that cannot be embedded. Further we provide an example of a planar graph that cannot be embedded together with a path when vertex correspondences are given.