Representing Directed Trees as Straight Skeletons
The straight skeleton of a polygon is the geometric graph obtained by tracing the vertices during a mitered offsetting process. It is known that the straight skeleton of a simple polygon is a tree, and one can naturally derive directions on the edges of the tree from the propagation of the shrinking process.
In this paper, we ask the reverse question: Given a tree with directed edges, can it be the straight skeleton of a polygon? And if so, can we find a suitable simple polygon? We answer these questions for all directed trees where the order of edges around each node is fixed.
- 1.Aichholzer, O., Aurenhammer, F.: Straight skeletons for general polygonal figures in the plane. In: Samoilenko, A. (ed.) Voronoi’s Impact on Modern Sciences II, vol. 21, pp. 7–21. Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev, Ukraine (1998)Google Scholar
- 2.Aichholzer, O., Aurenhammer, F., Alberts, D., Gärtner, B.: A novel type of skeleton for polygons. J. Univ. Comput. Sci. 1(12), 752–761 (1995)Google Scholar
- 3.Aichholzer, O., Biedl, T., Hackl, T., Held, M., Huber, S., Palfrader, P., Vogtenhuber, B.: Representing directed trees as straight skeletons [cs.CG] (2015). http://arxiv.org/abs/1508.01076
- 4.Aichholzer, O., Cheng, H., Devadoss, S.L., Hackl, T., Huber, S., Li, B., Risteski, A.: What makes a tree a straight skeleton? In: Proceedings of the 24th Canadian Conference on Computational Geometry, (CCCG 2012), pp. 253–258. Charlottetown, PE, Canada (2012)Google Scholar
- 5.Biedl, T., Held, M., Huber, S.: Recognizing straight skeletons and Voronoi diagrams and reconstructing their input. In: Gavrilova, M., Vyatkina, K. (eds.) Proceedings of the 10th International Symposium on Voronoi Diagrams in Science & Engineering (ISVD 2013), pp. 37–46. IEEE Computer Society, Saint Petersburg, Russia (2013)Google Scholar
- 6.Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane (Extended Abstract). In: Proceedings of 41st Annual ACM Symposium Theory Computing (STOC 2009), pp. 631–638. ACM, Bethesda, MD, USA (2009)Google Scholar