The Complexity of Simultaneous Geometric Graph Embedding

The Complexity of Simultaneous Geometric Graph Embedding

Given a collection of planar graphs G1, . . . , Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply k-SGE, is to find a set P of n points in the plane and a bijection φ : V → P such that the induced straight-line drawings of G1, . . . , Gk under φ are all plane. This problem is polynomial-time equivalent to weak rectilinear realizability of...

On Simultaneous Graph Embedding

We consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another where the mapping is not given. In particular, we show that without mapping, any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar gra...

Simultaneous Geometric Graph Embeddings

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...

Graph Simultaneous Embedding Tool, GraphSET

Problems in simultaneous graph drawing involve the layout of several graphs on a shared vertex set. This paper describes a Graph Simultaneous Embedding Tool, GraphSET, designed to allow the investigation of a wide range of embedding problems. GraphSET can be used in the study of several variants of simultaneous embedding including simultaneous geometric embedding, simultaneous embedding with fi...

Planar and Quasi Planar Simultaneous Geometric Embedding