Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

Abstract. A set of planar graphs share a simultaneous embedding if they can be drawn on the same vertex set V in the Euclidean plane without crossings between edges of the same graph. Fixed edges are common edges between graphs that share the same simple curve in the simultaneous drawing. Determining in polynomial time which pairs of graphs share a simultaneous embedding with fixed edges (SEFE) has been open. We give a necessary and sufficient condition for whether a SEFE exists for pairs of graphs whose union is homeomorphic to K5 or K3,3. This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide efficient algorithms to compute a SEFE. Finally, we provide a linear-time decision algorithm for deciding whether a pair of biconnected outerplanar graphs has a SEFE.

Characterizations of restricted pairs of planar graphs allowing simultaneous embedding with fixed edges

A set of planar graphs {G1(V,E1),…,Gk(V,Ek)} admits a simultaneous embedding if they can be drawn on the same pointset P of order n in the Euclidean plane such that each point in P corresponds one-to-one to a vertex in V and each edge in Ei does not cross any other edge in Ei (except at endpoints) for i∈{1,…,k}. A fixed edge is an edge (u,v) that is drawn using the same simple curve for each graph Gi whose edge set Ei contains the edge (u,v). We give a necessary and sufficient condition for two graphs whose union is homeomorphic to K5 or K3,3 to admit a simultaneous embedding with fixed edges (SEFE). This allows us to characterize the class of planar graphs that always have a SEFE with any other planar graph. We also characterize the class of biconnected outerplanar graphs that always have a SEFE with any other outerplanar graph. In both cases, we provide O(n4)-time algorithms to compute a SEFE

On a Tree and a Path with no Geometric Simultaneous Embedding

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: P -> V that induce planar straight-line embeddings both for $G_1$ and for $G_2$. While it is known that two caterpillars always admit a geometric simultaneous embedding and that two trees not always admit one, the question about a tree and a path is still open and is often regarded as the most prominent open problem in this area. 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. As a final result, we study the same problem when some constraints on the tree are imposed. Namely, we show that a tree of depth 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 depth 4.Comment: 42 pages, 33 figure