On a tree and a path with no geometric simultaneous embedding

Authors:
Patrizio Angelini;Markus Geyer;Michael Kaufmann;Daniel Neuwirth
Affiliations:
Dipartimento di Informatica e Automazione, Università Roma Tre, Italy;Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany;Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany;Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany
Venue:
GD'10 Proceedings of the 18th international conference on Graph drawing
Year:
2010

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Abstract

Two graphs G1 = (V,E1) and G2 = (V,E2) admit a geometric simultaneous embedding if there exists a set of points P and a bijection M : P → V that induce planar straight-line embeddings both for G1 and for G2. 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.