Article

Representing Directed Trees as Straight Skeletons

Authors:

Birgit VogtenhuberAuthors Info & Claims

Graph Drawing and Network Visualization

Pages 335 - 347

https://doi.org/10.1007/978-3-319-27261-0_28

Published: 24 September 2015 Publication History

Abstract

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.

References

[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

Google Scholar

[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

[7]

Di Battista, G., Lenhart, W., Liotta, G.: Proximity drawability: a survey. In: Tamassia, R., Tollis, I.G. (eds.) GD ’94. LNCS, vol. 894, pp. 328–339. Springer, Princeton, NJ, USA (1995)

Google Scholar

[8]

Dillencourt, M.B., Smith, W.D.: Graph-theoretical conditions for inscribability and delaunay realizability. Discrete Math. 161(1–3), 63–77 (1996)

Digital Library

Google Scholar

[9]

Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions. Discrete Comput. Geom. 22(4), 569–592 (1999)

Crossref

Google Scholar

[10]

Huber, S., Held, M.: A fast straight-skeleton algorithm based on generalized motorcycle graphs. Int. J. Comput. Geom. Appl. 22(5), 471–498 (2012)

Crossref

Google Scholar

[11]

Liotta, G., Lubiw, A., Meijer, H., Whitesides, S.: The rectangle of influence drawability problem. Comput. Geom. 10(1), 1–22 (1998)

Crossref

Google Scholar

[12]

Liotta, G., Meijer, H.: Voronoi drawings of trees. Comput. Geom. 24(3), 147–178 (2003)

Digital Library

Google Scholar

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Eder GHeld Mde Lorenzo SPalfrader P(2023)On the recognition and reconstruction of weighted Voronoi diagrams and bisector graphsComputational Geometry: Theory and Applications10.1016/j.comgeo.2022.101935109:COnline publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1016/j.comgeo.2022.101935

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Representing Directed Trees as Straight Skeletons

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Published In

Graph Drawing and Network Visualization

563 pages

ISBN:978-3-319-27260-3

DOI:10.1007/978-3-319-27261-0

Editors:
Emilio Di Giacomo
Università degli Studi di Perugia, Perugia, Italy
,
Anna Lubiw
School of Computer Science, University of Waterloo, WATERLOO, Canada

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Springer-Verlag

Berlin, Heidelberg

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Published: 24 September 2015

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Eder GHeld Mde Lorenzo SPalfrader P(2023)On the recognition and reconstruction of weighted Voronoi diagrams and bisector graphsComputational Geometry: Theory and Applications10.1016/j.comgeo.2022.101935109:COnline publication date: 1-Feb-2023
https://dl.acm.org/doi/10.1016/j.comgeo.2022.101935

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Decomposing a simple polygon into pseudo-triangles and convex polygons

A linear-time algorithm for constructing a circular visibility diagram

An Optimal Algorithm for Determining the Visibility of a Polygon from an Edge

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