Colored Simultaneous Geometric Embeddings
نویسندگان
چکیده
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 paths and (2) a 2-colored outerplanar graph together with any number of 2-colored paths. We also show that there does not exist a universal pointset of size n for paths colored with five colors. We finally show that the following simultaneous embeddings are not possible: (1) three 6-colored cycles, (2) four 6-colored paths, and (3) three 9-colored paths.
منابع مشابه
Constrained 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 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 کاملMonotone Simultaneous Embeddings of Paths in R^d
We study the following problem: Given k paths that share the same vertex set, is there a simultaneous geometric embedding of these paths such that each individual drawing is monotone in some direction? We prove that for any dimension d > 2, there is a set of d+ 1 paths that does not admit a monotone simultaneous geometric embedding.
متن کاملPoint-Set Embeddability of 2-Colored Trees
In this paper we study bichromatic point-set embeddings of 2-colored trees on 2-colored point sets, i.e., point-set embeddings of trees (whose vertices are colored red and blue) on point sets (whose points are colored red and blue) such that each red (blue) vertex is mapped to a red (resp. blue) point. We prove that deciding whether a given 2-colored tree admits a bichromatic point-set embeddin...
متن کامل