Planar Lombardi Drawings of Outerpaths
A Lombardi drawing of a graph is a drawingwhere edges are represented by circular arcs that meet at each vertex v with perfect angular resolution 360°/deg(v) . It is known that Lombardi drawings do not always exist, and in particular, that planar Lombardi drawings of planar graphs do not always exist , even when the embedding is not fixed. Existence of planar Lombardi drawings is known for restricted classes of graphs, such as subcubic planar graphs , trees , Halin graphs and some very symmetric planar graphs . On the other hand, all 2-degenerate graphs, including all outerplanar graphs, have Lombardi drawings, but not necessarily planar ones . One question that was left open is whether outerplanar graphs always have planar Lombardi drawings or not.
In this note, we report that the answer is “yes” for a more restricted subclass: the outerpaths, i.e., outerplanar graphs whose weak dual is a path. We sketch an algorithm that produces an outerplanar Lombardi drawing of any outerpath, in linear time.
- 1.Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Löffler, M.: Planar and Poly-arc Lombardi Drawings. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 308–319. Springer, Heidelberg (2012)Google Scholar
- 4.Eppstein, D.: Planar Lombardi Drawings for Subcubic Graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 126–137. Springer, Heidelberg (2013) To appear arXiv:1206.6142 Google Scholar