Trees on Tracks

Simultaneous embedding of planar graphs is related to the problems of graph thickness and geometric thickness. Techniques for simultaneous embedding of cycles have been used to show that the degree-4 graphs have geometric thickness at most two [3]. Simultaneous embedding techniques are also useful in visualization of graphs that evolve through time. The notion of simultaneous embedding is related to that of graph thickness. Two vertex-labeled planar graphs on n vertices can be simultaneously embedded if there exist a labeled point set of size n such that each of the graphs can be realized on that point set (using the vertexpoint mapping defined by the labels) with straight-line edge segments and without crossings. For example, any two paths can be simultaneously embedded, while there exist pairs of outerplanar graphs that do not have a simultaneous embedding. In this paper we present new results about embedding labeled trees and outerplanar graphs on labeled tracks, as well as related results on simultaneous embedding of tree-path pairs. In particular, we show that labeled trees cannot be embedded on labeled parallel straight-line tracks, but they can be embedded on labeled concentric circular tracks; see Fig. 1. The results generalize to outerplanar graphs as well. We also show that tree-path pairs can be simultaneously embedded when edges of the path are represented by circular arcs. Finally, we show how to embed a straight-line tree and a path with O(log n)-bends per edge, where n is the number of vertices.