Star-Shaped Drawings of Graphs with Fixed Embedding and Concave Corner Constraints
A star-shaped drawing of a graph is a straight-line drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. In this paper, given a biconnected planar graph G with fixed plane embedding and a subset A of corners of G, we consider the problem of finding a star-shaped drawing D of G such that only corners in A are allowed to become concave corners in D. We first characterize a necessary and sufficient condition for a subset A of corners to admit such a star-shaped drawing D. Then we present a linear time algorithm for finding such a star-shaped drawing D. Our characterization includes Thomassen’s classical characterization of biconnected plane graphs with a prescribed boundary that have convex drawings.
Unable to display preview. Download preview PDF.
- 1.Chiba, N., Yamanouchi, T., Nishizeki, T.: Linear Algorithms for Convex Drawings of Planar Graphs. In: Progress in Graph Theory, pp. 153–173. Academic Press, London (1984)Google Scholar
- 2.Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1998)Google Scholar
- 4.Fáry, I.: On Straight Line Representations of Planar Graphs. Acta Sci. Math. Szeged 11, 229–233 (1948)Google Scholar
- 6.Hong, S.-H., Nagamochi, H.: Convex Drawings of Hierarchical Plane Graphs. In: Proc. of AWOCA 2006 (2006)Google Scholar
- 7.Hong, S.-H., Nagamochi, H.: Fully Convex Drawings of Clustered Planar Graphs. In: Proc. of WAAC 2007, pp. 32–39 (2007)Google Scholar
- 8.Hong, S.-H., Nagamochi, H.: Star-Shaped Drawings of Planar Graphs. In: Proc. of IWOCA (to appear, 2007)Google Scholar
- 9.Hong, S.-H., Nagamochi, H.: Star-Shaped Drawings of Graphs with Fixed Embedding and Concave Corner Constraints. Technical Report TR 2008-002, Department of Applied Mathematics and Physics, Kyoto University (2008)Google Scholar
- 10.Kant, G.: Algorithms for Drawing Planar Graphs. Ph.D. Dissertation, Department of Computer Science, University of Utrecht, Holland (1993)Google Scholar
- 12.Thomassen, C.: Plane Representations of Graphs. In: Progress in Graph Theory, pp. 43–69. Academic Press, London (1984)Google Scholar