Journal of Graph Algorithms and Applications
http://jgaa.info ISSN: 1526-1719
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DOI: 10.7155/jgaa.00250 On a Tree and a Path with no Geometric Simultaneous Embedding Patrizio Angelini , Markus Geyer , Michael Kaufmann , and Daniel Neuwirth Vol. 16, no. 1, pp. 37-83, 2012. Regular paper Abstract Two graphs G1=(V,E1) and G2=(V,E2) admit a geometric simultaneous embedding if there exist a set of points P and a bijection M: V→ P that induce planar straight-line embeddings both for G1 and for G2. The most prominent problem in this area is the question of whether a tree and a path can always be simultaneously embedded. We answer this question in the negative by providing a counterexample. Additionally, since the counterexample uses disjoint edge sets for the two graphs, we also negatively answer another open question, that is, whether it is possible to simultaneously embed two edge-disjoint trees. Finally, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of height 2 and a path always admit a geometric simultaneous embedding. In fact, such a strong constraint is not so far from closing the gap with the instances not admitting any solution, as the tree used in our counterexample has height 4.
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Submitted: December 2010. Reviewed: April 2011. Reviewed: October 2011. Accepted: November 2011. Final: November 2011. Revised: November 2011. Published: January 2012. Revised: August 2011. Communicated by Ulrik Brandes and Sabine Cornelsen |
| Journal of Graph Algorithms and Applications |