Abstract
In this paper we establish four necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient. We present an 0(n2)-time algorithm for testing the first and second necessary conditions and leave it open whether the third and fourth necessary conditions can be tested in polynomial time. We also show that visibility graphs of simple polygons do not possess the characteristics of a few special classes of graphs
Article PDF
Similar content being viewed by others
References
J. Abello, O. Egecioglu, and K. Kumar, Visibility graphs of staircase polygons and the weak Bruhat order, I: from visibility graphs to maximal chains,Discrete & Computational Geometry, 14(3) (1995), 331–358.
J. Abello, O. Egecioglu, and K. Kumar, Visibility graphs of staircase polygons and the weak Bruhat order, II: from maximal chains to polygons, preprint.
J. Abello and K. Kumar, Visibility graphs and oriented metroids,Proceeding of Graph Drawing, Lecture Notes in Computer Science, Vol.894, Springer-Verlag, Berlin, pp. 147–158, 1995.
J. Abello, H. Lin, and S. Pisupati, On visibility graphs of simple polygons,Congressus Numeratium, 90 (1992), 119–128.
D. Avis and D. Rappaport, Computing the largest empty convex subset of a set of points,Proceedings of the First ACM Symposium on Computational Geometry, pp. 161-167, 1985.
M. A. Buckinghan, Circle Graphs, Ph.D. Dissertation, Report No. NSO-21, Courant Institute of Mathematical Sciences, New York, 1980.
H. ElGindy, Hierarchical decomposition of polygons with applications, Ph.D. Dissertation, McGill University, Montreal, 1985.
H. Everett, Visibility graph recognition, Ph.D. Dissertation, University of Toronto, Toronto, January 1990.
H. Everett and D. Corneil, Recognizing visibility graphs of spiral polygons,Journal of Algorithms, 11 (1990), 1–26.
C. P. Gabor, W. Hsu, and K. J. Supowit, Recognizing circle graphs in polynomial time,Proceedings of the 26th IEEE Annual Symposium on Foundation of Computer Science, pp. 106-116, 1985.
F. Gravil, Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph,SIAM Journal on Computing, 1 (1972), 180–187.
S. K. Ghosh, On recognizing and characterizing visibility graphs of simple polygons, Report JHU/EECS-86/14, The Johns Hopkins University, Baltimore, 1986. Also inProceedings of the Scandinavian Workshop on Algorithm Theory, Lecture Notes in Computer Science, Vol.318, Springer-Verlag, Berlin, pp. 96–104, 1988.
S. K. Ghosh, A. Maheshwari, S. P. Pal, S. Saluja, and C. E. Veni Madhavan, Characterizing and recognizing weak visibility polygons,Computational Geometry: Theory and Applications, 3 (1993), 213–233.
M. C. Golumbic,Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
J. Hershberger, An optimal visibility graph algorithm for triangulated simple polygon,Algorithmica, 4 (1989), 141–155.
T. Lazano-Perez and M. A. Wesley, An algorithm for planning collision free paths among polygonal obstacles,Communications of the ACM, 22 (1979), 560–570.
J. O’Rourke,Art Gallery Theorems and Algorithms, Oxford University Press, Oxford, 1987.
J. O’Rourke, Computational geometry column 18,SIGACT News, 24 (1993), 20–25.
L. G. Shapiro and R. M. Haralick, Decomposition of two-dimensional shape by graph-theoretic clustering,IEEE Transactions on Pattern Analysis and Machine Intelligence, 1 (1979), 10–19.
T. Shermer, Hiding people in polygons,Computing, 42 (1989), 109–132.
G. Srinivasaraghavan and A. Mukhopadhyay, A new necessary condition for the vertex visibility graphs of simple polygons,Discrete & Computational Geometry, 12 (1994), 65–82.
Author information
Authors and Affiliations
Corresponding author
Additional information
Part of this work was done when the author visited the John Hopkins University and was Supported by NSF Grant DCR83-51468 and a grant from IBM.
Rights and permissions
About this article
Cite this article
Ghosh, S.K. On recognizing and characterizing visibility graphs of simple polygons. Discrete Comput Geom 17, 143–162 (1997). https://doi.org/10.1007/BF02770871
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02770871