Colored Simultaneous Geometric Embeddings and Universal Pointsets

Abstract

Universal pointsets can be used for visualizing multiple relationships on the same set of objects or for visualizing dynamic graph processes. In simultaneous geometric embeddings, the same point in the plane is used to represent the same object as a way to preserve the viewer's mental map. In colored simultaneous embeddings this restriction is relaxed, by allowing a given object to map to a subset of points in the plane. Specifically, consider a set of graphs on the same set of n vertices partitioned into k colors. Finding a corresponding set of k-colored points in the plane such that each vertex is mapped to a point of the same color so as to allow a straightline plane drawing of each graph is the problem of colored simultaneous geometric embedding.Work on this paper began at 570 Algorithmica (2011) 60: 569-592 For n-vertex paths, we show that there exist universal pointsets of size n, colored with two or three colors. We use this result to construct colored simultaneous geometric embeddings for a 2-colored tree together with any number of 2-colored paths, and more generally, a 2-colored outerplanar graph together with any number of 2-colored paths. For n-vertex trees, we construct small near-universal pointsets for 3-colored caterpillars of size n, 3-colored radius-2 stars of size n + 3, and 2-colored spiders of size n. For n-vertex outerplanar graphs, we show that these same universal pointsets also suffice for 3-colored K 3 -caterpillars, 3-colored K 3 -stars, and 2-colored fans, respectively. We also present several negative results, showing that there exist a 2-colored planar graph and pseudo-forest, three 3-colored outerplanar graphs, four 4-colored pseudo-forests, three 5-colored pseudo-forests, five 5-colored paths, two 6-colored biconnected outerplanar graphs, three 6-colored cycles, four 6-colored paths, and three 9-colored paths that cannot be simultaneously embedded.

Universal Sets for Straight-Line Embeddings of Bicolored Graphs

“…A k-colored set S of n points is universal for a class G of graphs if every (not necessarily proper) coloring of vertices of G ∈ G on n vertices admits an embedding on S. Brandes et al [4] find, for example, universal k-colored point sets for the class of caterpillars for every k ≤ 3.…”

Simultaneous Embedding of Colored Graphs

“…Colored simultaneous embedding was first considered by Brandes et al [2], where they examined classes of graphs that admit simultaneous embedding with bend complexity 0, i.e., each edge is a straight line segment, which is also known as simultaneous geometric embedding. Although every pair of paths admits simultaneous geometric embedding, there exist 3 paths that do not admit simultaneous embedding [3], even when colored with 6 colors [2]. However, any number of 3-colored paths can be simultaneously Here the vertices are colored with red, blue, and black (i.e., large, small, and tiny discs).…”

Planar Drawings of Fixed-Mobile Bigraphs

“…Point-set embedding. In a point-set embedding problem, a planar graph with n vertices must be planarly mapped onto a given set of n points, with or without a predefined correspondence between the vertices and the points (see, e.g., [1,8,11,12,22,30]). Thus, in all settings of the point-set embedding problem, each vertex can only be mapped to a finite set of points.…”

“…We remark that in a preliminary version of[31], it is claimed membership in NP for the partial planarity extension problem[32], which would imply membership in NP also for our problem. That claim, however, lacks a proof in[32] and the author was only able to prove the NP-hardness of the problem in[31] (personal communication) 8. Cells in the intersection of two convex hulls correspond to different vertices of Gc.…”