Two trees which are self-intersecting when drawn simultaneously

Abstract

A current topic in graph drawing is the question how to draw two edge sets on the same vertex set, the so-called simultaneous drawing of graphs. The goal is to simultaneously find a nice drawing for both of the sets. It has been found out that only restricted classes of planar graphs can be drawn simultaneously using straight lines and without crossings within the same edge set. In this paper, we negatively answer one of the most often posted open questions namely whether any two trees with the same vertex set can be drawn simultaneously crossing-free in a straight-line way.

Matched Drawings of Planar Graphs

“…These results show that matched drawings do indeed allow larger classes of graphs to be drawn than simultaneous geometric embeddings with mapping (a path and a planar graph may not admit a simultaneous geometric embedding with mapping [3] and the same negative result also holds for pairs of trees [15]). …”

“…Brass et al [3] showed how to simultaneously embed pairs of paths, pairs of cycles, and pairs of caterpillars, but they also proved that a path and a graph or two outerplanar graphs may not admit this type of drawing. Geyer, Kaufmann, and Vrt'o [15] recently proved that even a pair of trees may not have a simultaneous geometric embedding with mapping. These negative results motivated the study of relaxations of simultaneous geometric embeddings.…”

Characterization of Unlabeled Level Planar Graphs

“…While Geyer et al [7] have shown this cannot always be done for tree-tree pairs, the question remains open for tree-path pairs. Estrella et al [5] partially answer this question by characterizing the set of trees that have a simultaneous geometric embedding with a strictly monotone path.…”

Embedding Graphs Simultaneously with Fixed Edges

“…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.…”