GD 2010: Graph Drawing pp 38-49

# On a Tree and a Path with No Geometric Simultaneous Embedding

• Patrizio Angelini
• Markus Geyer
• Michael Kaufmann
• Daniel Neuwirth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)

## Abstract

Two graphs G 1 = (V,E 1) and G 2 = (V,E 2) admit a geometric simultaneous embedding if there exists a set of points P and a bijection M : PV  that induce planar straight-line embeddings both for G 1 and for G 2. The most prominent problem in this area is the question 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 prove that it is not always 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.

## Keywords

Planar Graph Extended Formation Outerplanar Graph Planar Embedding Planar Drawing
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Patrizio Angelini
• 1
• Markus Geyer
• 2
• Michael Kaufmann
• 2
• Daniel Neuwirth
• 2
1. 1.Dipartimento di Informatica e AutomazioneUniversità Roma TreItaly
2. 2.Wilhelm-Schickard-Institut für InformatikUniversität TübingenGermany