Outerplanar Graphs and Trees on Tracks

Given a vertex-labeled tree on n vertices we show how to obtain a straight-line, crossings-free drawing of it on a set of n labeled concentric tracks, such that the vertex labels match the track labels. The tracks can be defined by conic sections (such as circles, ellipses, circular arcs) or other smooth convex curves. We show that this type of embedding can be used to simultaneously embed tree-path pairs, such that the tree is drawn without crossings, using one straight-line segment per edge, and the path is drawn without crossings, using one circular arc segment per edge. This result generalizes to outerplanar graphs. We also consider star-track embeddings of trees which we use to obtain simultaneous embeddings of tree-path pairs using piecewise linear edges. In particular, we show how to simultaneously embed tree-path pairs so that the tree is drawn without crossings, using one straight-line segment per edge and the path is drawn without crossings, using at most 2 bends per edge. These results also generalize to outerplanar graphs.