A pair of trees without a simultaneous geometric embedding in the plane
<|reference_start|>A pair of trees without a simultaneous geometric embedding in the plane: 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.<|reference_end|>
@article{kutz2005a,
    title={A pair of trees without a simultaneous geometric embedding in the plane},
    author={Martin Kutz},
    journal={arXiv preprint arXiv:cs/0510053},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0510053},
    primaryClass={cs.CG}
}