Characterizing and recognizing visibility graphs of Funnel-shaped polygons
A funnel, which is notable for its fundamental role in visibility algorithms, is defined as a polygon that has exactly three convex vertices two of which are connected by a boundary edge. In this paper, we investigate the visibility graph of a funnel which we call an F-graph. We first present two characterizations of an F-graph, one of whose sufficiency proof itself is an algorithm to draw a corresponding funnel on the plane in O(e) time, where e is the number of the edges in an input graph. We next give an O(e) time algorithm for recognizing an F-graph. When the algorithm recognizes a graph to be an F-graph, it also reports one of the Hamiltonian cycles defining the boundary of a corresponding funnel. We finally show that an F-graph is weakly triangulated and therefore perfect. This agrees with the fact that many of perfect graphs are related to geometric structures.
Unable to display preview. Download preview PDF.
- 1.Abello, J., O. Egecioglu, and K. Kumar, “Characterizing visibility graphs of staircase polygons,” manuscript (extended abstract), 1991.Google Scholar
- 2.Asano, T., T. Asano, L. Guibas, J. Hershberger, and H. Imai, “Visibility of disjoint polygons,” Algorithmica, vol. 1, pp. 49–63, 1986.Google Scholar
- 3.Avis, D. and G. T. Toussaint, “An optimal algorithm for determining the visibility of a polygon from an edge,” IEEE Trans. Computers, vol. 30, no. 12, pp. 910–914, 1981.Google Scholar
- 4.Chazelle, B. and L. Guibas, “Visibility and intersection problems in plane geometry,” Discrete and Computational Geometry, vol. 4, pp. 551–581, 1989.Google Scholar
- 5.Choi, S.-H., S. Y. Shin, and K.-Y. Chwa, “Characterizing and recognizing the visibility graph of a funnel-shaped polygon,” Tech. Report CS-TR-92-71, Dept. Comp. Sci., KAIST, Korea, 1992.Google Scholar
- 6.Coullard, C. and A. Lubiw, “Distance visibility graphs,” Proc. 7th ACM Symp. Computational Geometry, pp. 289–296, 1991.Google Scholar
- 7.ElGindy, H. A., “Hierarchical decomposition of polygons with applications,” Ph.D. dissertation, School of Comp. Sci., McGill Univ., Montreal, 1985.Google Scholar
- 9.Ghosh, S. K., “On recognizing and characterizing visibility graph of simple polygons,” Tech. Report JHU/EECS-86/14, Dept. EECS, The Johns Hopkins Univ., Baltimore, 1986. Also in Lecture Notes in Computer Science 318, R. Karlsson and A. Lingas Eds., Springer-Verlag, New York, 1988Google Scholar
- 11.Golumbic, M. C., Algorithmic graph theory and perfect graphs, Academic Press, New York, 1980.Google Scholar
- 13.Hayward, R. B., “Weakly triangulated graphs,” J. Combinatorial Theory, B, vol. 39, pp. 200–209, 1985.Google Scholar
- 16.Lee, D. T. and F. P. Preparata, “Euclidean shortest paths in the presence of rectilinear barriers,” Networks, vol. 14, no. 3, pp. 393–410, 1984.Google Scholar
- 17.O'Rourke, J., Art gallery theorems and algorithms, Oxford Univ. Press, New York, 1987.Google Scholar