On a Tree and a Path with no Geometric Simultaneous Embedding

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. Submitted: December 2010 Reviewed: April 2011 Revised: August 2011 Reviewed: October 2011 Revised: November 2011 Accepted: November 2011 Final: November 2011 Published: January 2012 Article type: Regular paper Communicated by: U. Brandes and S. Cornelsen Research partially supported by MIUR (Italy), Project AlgoDEEP no. 2008TFBWL4, and by the ESF, Project 10-EuroGIGA-OP-003 GraDR “Graph Drawings and Representations. E-mail addresses: [email protected] (Patrizio Angelini) [email protected] (Markus Geyer) [email protected] (Michael Kaufmann) [email protected] (Daniel Neuwirth) 38 Angelini, Geyer, Kaufmann, Neuwirth Tree-Path Counterexample