GD 2001: Graph Drawing pp 407-421

Polar Coordinate Drawing of Planar Graphs with Good Angular Resolution

• Duncan Christian A.
• Kobourov Stephen G.
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)

Abstract

We present a novel way to draw planar graphs with good angular resolution. We introduce the polar coordinate representation and describe a family of algorithms which use polar representation. The main advantage of using a polar representation is that it allows us to exert independent control over grid size and bend positions. Polar coordinates allow us to specify different vertex resolution, bend-point resolution and edge separation. We first describe a standard (Cartesian) representation algorithm (CRA) which we then modify to obtain a polar representation algorithm (PRA). In both algorithms we are concerned with the following drawing criteria: angular resolution, bends per edge, vertex resolution, bend-point resolution, edge separation, and drawing area. The CRA algorithm achieves 1 bend per edge, unit vertex and bend resolution, $$\sqrt 2 /2$$ edge separation, $$5n \times \tfrac{{5n}} {2}$$ drawing area and $$\tfrac{1} {{2d(\upsilon )}}$$ angular resolution, where d(v) is the degree of vertex v. The PRA algorithm has an improved angular resolution of $$\tfrac{\pi } {{4d(\upsilon )}}$$ , 1 bend per edge, and unit vertex resolution. For the PRA algorithm, the bend-point resolution and edge separation are parameters that can be modified to achieve different types of drawings and drawing areas. In particular, for the same parameters as the CRA algorithm (unit bend-point resolution and $$\sqrt 2 /2$$ edge separation), the PRA algorithm creates a drawing of size $$9n \times \tfrac{{9n}} {2}$$.

References

1. 1.
C. C. Cheng, C. A. Duncan, M. T. Goodrich, and S. G. Kobourov. Drawing planar graphs with circular arcs. Discrete and Computational Geometry, 25:405–418, 2001.
2. 2.
M. Chrobak and T. Payne. A linear-time algorithm for drawing planar graphs. Inform. Process. Lett., 54:241–246, 1995.
3. 3.
H. de Fraysseix, J. Pach, and R. Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41–51, 1990.
4. 4.
G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, Englewood Cliffs, NJ, 1999.
5. 5.
A. Garg and R. Tamassia. Planar drawings and angular resolution: Algorithms and bounds. In Proc. 2nd European Symposium on Algorithms, pages 12–23, 1994.Google Scholar
6. 6.
M. T. Goodrich and C. G. Wagner. A framework for drawing planar graphs with curves and polylines. In Proc. 6th Symposium on Graph Drawing, pages 153–166, 1998.Google Scholar
7. 7.
C. Gutwenger and P. Mutzel. Planar polyline drawings with good angular resolution. In Proc. 6th Symposium on Graph Drawing, pages 167–182, 1998.Google Scholar
8. 8.
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).Google Scholar
9. 9.
S. Malitzand A. Papakostas. On the angular resolution of planar graphs. SIAM J. Discrete Math., 7:172–183, 1994.