Abstract
An open edge of a simple polygon is the set of points in the relative interior of an edge. We revisit several art gallery problems, previously considered for closed edge guards, using open edge guards. A guard edge of a polygon is an edge that sees every point inside the polygon. We show that every simple non-starshaped polygon admits at most one open guard edge, and give a simple new proof that it admits at most three closed guard edges. We also characterize open guard edges using a special type of kernel. Finally, we present lower bound constructions for simple polygons with n vertices that require \(\lfloor n/3 \rfloor\) open edge guards, and conjecture that this bound is tight.
Dedicated to Ferran Hurtado on the occasion of his 60th birthday.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Avis, D., Gum, T., Toussaint, G.T.: Visibility between two edges of a simple polygon. The Visual Computer 2, 342–357 (1986)
Avis, D., Toussaint, G.: An optimal algorithm for determining the visibility of a polygon from an edge. IEEE Tran. Comput. C-30, 910–914 (1981)
Bhattacharya, B.K., Das, G., Mukhopadhyay, A., Narasimhan, G.: Optimally computing a shortest weakly visible line segment inside a simple polygon. Comput. Geom. Theory Appl. 23, 1–29 (2002)
Chen, D.Z.: Optimally computing the shortest weakly visible subedge of a simple polygon. J. Algorithms 20, 459–478 (1996)
Chvátal, V.: A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B 28, 39–41 (1975)
Lee, D.T., Preparata, F.: An optimal algorithm for finding the kernel of a polygon. J. ACM 26, 415–421 (1979)
Lu, B.-K., Hsu, F.-R., Tang, C.Y.: Finding the shortest boundary guard of a simple polygon. Theor. Comp. Sci. 263, 113–121 (2001)
O’Rourke, J.: Galleries need fewer mobile guards: A variation on Chvátal’s theorem. Geometriae Dedicata 14, 273–283 (1983)
Park, J., Shin, S.Y., Chwa, K., Woo, T.C.: On the number of guard edges of a polygon. Discrete Comput. Geom. 10, 447–462 (1993)
Shermer, T.C.: Recent results in art galleries. Proc. IEEE 80, 1384–1399 (1992)
Shermer, T.C.: A tight bound on the combinatorial edge guarding problem. Snapshots of Comp. and Discrete Geom. 3, 191–223 (1994)
Shin, S.Y., Woo, T.: An optimal algorithm for finding all visible edges in a simple polygon. IEEE Tran. Robotics and Automation 5, 202–207 (1989)
Tan, X.: Fast computation of shortest watchman routes in simple polygons. Inf. Proc. Lett. 77, 27–33 (2001)
Viglietta, G.: Searching polyhedra by rotating planes, manuscript, arXiv:1104.4137 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Tóth, C.D., Toussaint, G.T., Winslow, A. (2012). Open Guard Edges and Edge Guards in Simple Polygons. In: Márquez, A., Ramos, P., Urrutia, J. (eds) Computational Geometry. EGC 2011. Lecture Notes in Computer Science, vol 7579. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34191-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-34191-5_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34190-8
Online ISBN: 978-3-642-34191-5
eBook Packages: Computer ScienceComputer Science (R0)