Simultaneous Geometric Graph Embeddings
نویسندگان
چکیده
We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straight-line drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NP-hard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a longstanding open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE.
منابع مشابه
Colored Simultaneous Geometric Embeddings
We introduce the concept of colored simultaneous geometric embeddings as a generalization of simultaneous graph embeddings with and without mapping. We show that there exists a universal pointset of size n for paths colored with two or three colors. We use these results to show that colored simultaneous geometric embeddings exist for: (1) a 2-colored tree together with any number of 2-colored p...
متن کاملGeometric Simultaneous Embeddings of a Graph and a Matching
The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we pr...
متن کاملConstrained Simultaneous and Near-Simultaneous Embeddings
A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ : V1 → V2 is a pair of planar straightline drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, where v1 ∈ V1 and v2 ∈ V2. In this paper we examine several constrained versions of the geometric si...
متن کاملA note on simultaneous embedding of planar graphs
Let G1 and G2 be a pair of planar graphs such that V (G1) = V (G2) = V . A simultaneous embedding [6] Ψ = (Γ1,Γ2) of G1 and G2 is a pair of crossing-free drawings Γ1 and Γ2 of G1 and G2, respectively, such that for every vertex v ∈ V we have Γ1(v) = Γ2(v). If every edge e ∈ E(G1) ∩ E(G2) is represented with the same simple open Jordan curve both in Γ1 and in Γ2 we say that Ψ is a simultaneous e...
متن کاملA pair of trees without a simultaneous geometric embedding in the plane
Any planar graph has a crossing-free straight-line drawing in the plane. A simultaneous geometric embedding of two n-vertex graphs is a straight-line drawing of both graphs on a common set of n points, such that the edges withing each individual graph do not cross. We consider simultaneous embeddings of two labeled trees, with predescribed vertex correspondences, and present an instance of such...
متن کامل