Drawing planar graphs with circular arcs
 C. C. Cheng,
 C. A. Duncan,
 M. T. Goodrich,
 S. G. Kobourov
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
In this paper we address the problem of drawing planar graphs with circular arcs while maintaining good angular resolution and small drawing area. We present a lower bound on the area of drawings in which edges are drawn using exactly one circular arc. We also give an algorithm for drawing nvertex planar graphs such that the edges are sequences of two continuous circular arcs. The algorithm runs in O(n) time and embeds the graph on the O(n) × O(n) grid, while maintaining Θ(1/d(v)) angular resolution, where d(v) is the degree of vertex v. Since in this case we use circular arcs of infinite radius, this is also the first algorithm that simultaneously achieves good angular resolution, small area, and at most one bend per edge using straightline segments. Finally, we show how to create drawings in which edges are smooth C ^{1}continuous curves, represented by a sequence of at most three circular arcs.
 G. Di Battista, P. Eades, R. Tamassia, and I. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. PrenticeHall, Englewood Cliffs, NJ, 1999.
 M. Chrobak and T. Payne. A lineartime algorithm for drawing planar graphs. Information Processing Letters, 54:241–246, 1995. CrossRef
 H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41–51, 1990. CrossRef
 I. Fáry. On straight lines representation of planar graphs. Acta Scientiarum Mathematicarum, 11:229–233, 1948.
 M. Formann, T. Hagerup, J. Haralambides, M. Kaufmann, F. T. Leighton, A. Simvonis, E. Welzl, and G. Woeginger. Drawing graphs in the plane with high resolution. SIAM Journal of Computing, 22:1035–1052, 1993. CrossRef
 A. Garg and R. Tamassia. Planar drawings and angular resolution: algorithms and bounds. In Proceedings of the 2nd Annual European Symposium on Algorithms, pages 12–23, 1994.
 M. T. Goodrich and C. G. Wagner. A framework for drawing planar graphs with curves and polylines. In Proceedings of the 6th Annual Symposium on Graph Drawing, pages 153–166, 1998.
 C. Gutwenger and P. Mutzel. Planar polyline drawings with good angular resolution. In Proceedings of the 6th Annual Symposium on Graph Drawing, pages 167–182, 1998.
 G. Kant. Drawing planar graphs using the lmcordering. In Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 101–110, 1992.
 G. Kant. Algorithms for Drawing Planar Graphs. Ph.D. thesis, Department of Computer Science, University of Utrecht, Utrecht, 1993.
 G. Kant. Drawing planar graphs using the canonical ordering. Algorithmica, 16: 4–32, 1996 (special issue on Graph Drawing, edited by G. Di Battista and R. Tamassia). CrossRef
 S. Malitz and A. Papakostas. On the angular resolution of planar graphs. SIAM Journal of Discrete Mathematics, 7:172–183, 1994. CrossRef
 W. Schnyder. Embedding planar graphs on the grid. In Proceedings of the 1st ACMSIAM Symposium on Discrete Algorithms (SODA), pages 138–148, 1990.
 W. T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 13(52):743–768, 1963. CrossRef
 K. Wagner. Bemerkungen zum vierfarbenproblem. Jahresbericht der Deutschen MathematikerVereinigung, 46:26–32, 1936.
 Title
 Drawing planar graphs with circular arcs
 Journal

Discrete & Computational Geometry
Volume 25, Issue 3 , pp 405418
 Cover Date
 20010401
 DOI
 10.1007/s004540010080
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 C. C. Cheng ^{(1)}
 C. A. Duncan ^{(2)}
 M. T. Goodrich ^{(3)}
 S. G. Kobourov ^{(4)}
 Author Affiliations

 1. Mathematical Sciences Department, Johns Hopkins University, 21218, Baltimore, MD, USA
 2. Department of Computer Science, University of Miami, 33124, Coral Gables, FL, USA
 3. Department of Computer Science, Johns Hopkins University, 21218, Baltimore, MD, USA
 4. Department of Computer Science, University of Arizona, 85721, Tucson, AZ, USA