On Simultaneous Planar Graph Embeddings

Abstract

Abstract. 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 in which the mapping is not given. In particular, given a mapping, we show how to embed two paths on an n × n grid, and two caterpillar graphs on a 3n × 3n grid. We show that it is not always possible to simultaneously embed three paths. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n) × O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n 2 ) × O(n 2 ) grid.

An Interactive Multi-user System for Simultaneous Graph Drawing

“…11 and Fig. 12 can be found in [2] and [6], respectively. However, there are many other examples that we have neither been able to realize simultaneously, nor prove that it cannot be done.…”

“…The problem of simultaneous embedding of planar graphs was introduced in [2], where it is shown that pairs of paths, cycles, and caterpillars can always be realized, while for general planar graphs and even outerplanar graphs this is not always possible. Modified force-directed methods are used to visualize general graphs simultaneously such that the mental map is preserved up in [7].…”

“…6-7. It is also known that there exist pairs of outerplanar graphs that cannot be realized simultaneously [2]; see Fig. 11.…”

Embedding Graphs Simultaneously with Fixed Edges

“…A simultaneous embedding of two planar graphs G 1 and G 2 is a pair of drawings of G 1 and G 2 such that each drawing is planar and each vertex common to G 1 and G 2 is represented by the same point in both drawings. Unfortunately, if one wishes to visualize the edges of G 1 and G 2 as rectilinear segments (the so called geometric simultaneous embedding), not all pairs of graphs can be embedded simultaneously.…”

“…Unfortunately, if one wishes to visualize the edges of G 1 and G 2 as rectilinear segments (the so called geometric simultaneous embedding), not all pairs of graphs can be embedded simultaneously. Erten and Kobourov ([4]), Brass et al ( [1]), and Geyer et al ( [7]) have shown that it is not always possible to embed simultaneously with straight-line edges a planar graph and a path, three paths, and two trees, respectively. On the other hand, if one permits that each edge of a graph is displayed as a different Jordan curve (the so called simultaneous embedding), then by the results of Pach and Wenger ( [9]) any number of planar graphs can be embedded simultaneously.…”