On a Tree and a Path with no Geometric Simultaneous Embedding

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.