GD 2006: Graph Drawing pp 138-149

# Open Rectangle-of-Influence Drawings of Inner Triangulated Plane Graphs

• Kazuyuki Miura
• Tetsuya Matsuno
• Takao Nishizeki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4372)

## Abstract

A straight-line drawing of a plane graph is called an open rectangle-of-influence drawing if there is no vertex in the proper inside of the axis-parallel rectangle defined by the two ends of every edge. In an inner triangulated plane graph, every inner face is a triangle although the outer face is not always a triangle. In this paper, we first obtain a sufficient condition for an inner triangulated plane graph G to have an open rectangle-of-influence drawing; the condition is expressed in terms of a labeling of angles of a subgraph of G. We then present an O(n 1.5/logn)-time algorithm to examine whether G satisfies the condition and, if so, construct an open rectangle-of-influence drawing of G on an (n − 1) ×(n − 1) integer grid, where n is the number of vertices in G.

### References

1. 1.
Biedl, T., Bretscher, A., Meijer, H.: Rectangle of influence drawings of graphs without filled 3-cycles. In: Kratochvíl, J. (ed.) GD 1999. LNCS, vol. 1731, pp. 359–368. Springer, Heidelberg (1999)
2. 2.
Biedl, T., Kant, G., Kaufmann, M.: On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997)
3. 3.
Chrobak, M., Kant, G.: Convex grid darwings of 3-connected planar graphs. Int. J. Comput. Geom Appl. 7(3), 211–223 (1997)
4. 4.
Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Upper Saddle River (1999)
5. 5.
Di Battista, G., Lenhart, W., Liotta, G.: Proximity drawability: A survey. In: Tamassia, R., Tollis, I(Y.) G. (eds.) GD 1994. LNCS, vol. 894, pp. 328–339. Springer, Heidelberg (1995)Google Scholar
6. 6.
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a graph on a grid. Combinatorica 10, 41–51 (1990)
7. 7.
8. 8.
Hochbaum, D.S.: Faster pseudoflow-based algorithms for the bipartite matching and the closure problems (Abstract). In: CORS/SCRO-INFORMS Joint Int. Meeting, Banff, Canada, May 16-19, p. 46 (2004)Google Scholar
9. 9.
Hochbaum, D.S., Chandran, B.G.: Further below the flow decomposition barrier of maximum flow for bipartite matching and maximum closure. Working paper (2004)Google Scholar
10. 10.
Kant, G., He, X.: Regular edge-labeling of 4-connected plane graphs and its applications in graph drawing problems. Theoretical Computer Science 172, 175–193 (1997)
11. 11.
Liotta, G., Lubiw, A., Meijer, H., Whitesides, S.H.: The rectangle of influence drawability problem. Comput. Geom. Theory and Applications 10(1), 1–22 (1998)
12. 12.
Miura, K., Nishizeki, T.: Rectangle-of-influence drawings of four-connected plane graphs. In: Proc. of Information Visualisation 2005, Asia-Pacific Symposium on Information Visualisation (APVIS2005). ACS, vol. 45, pp. 71–76 (2005)Google Scholar
13. 13.
Miura, K., Nakano, S., Nishizeki, T.: Grid drawings of four-connected plane graphs. Discrete & Computational Geometry 26(1), 73–87 (2001)
14. 14.
Nishizeki, T., Rahman, M.S.: Planar Graph Drawing. World Scientific, Singapore (2004)
15. 15.
Schnyder, W.: Embedding planar graphs in the grid. In: Proc. 1st Annual ACM-SIAM Symp. on Discrete Algorithms, San Francisco, pp. 138–147. ACM Press, New York (1990)Google Scholar