Open Guard Edges and Edge Guards in Simple Polygons

  • Csaba D. Tóth
  • Godfried T. Toussaint
  • Andrew Winslow
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


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.


art gallery illumination visibility mobile guards 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avis, D., Gum, T., Toussaint, G.T.: Visibility between two edges of a simple polygon. The Visual Computer 2, 342–357 (1986)CrossRefzbMATHGoogle Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, D.Z.: Optimally computing the shortest weakly visible subedge of a simple polygon. J. Algorithms 20, 459–478 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chvátal, V.: A combinatorial theorem in plane geometry. J. Combin. Theory Ser. B 28, 39–41 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lee, D.T., Preparata, F.: An optimal algorithm for finding the kernel of a polygon. J. ACM 26, 415–421 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    O’Rourke, J.: Galleries need fewer mobile guards: A variation on Chvátal’s theorem. Geometriae Dedicata 14, 273–283 (1983)MathSciNetzbMATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Shermer, T.C.: Recent results in art galleries. Proc. IEEE 80, 1384–1399 (1992)CrossRefGoogle Scholar
  11. 11.
    Shermer, T.C.: A tight bound on the combinatorial edge guarding problem. Snapshots of Comp. and Discrete Geom. 3, 191–223 (1994)Google Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Tan, X.: Fast computation of shortest watchman routes in simple polygons. Inf. Proc. Lett. 77, 27–33 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Viglietta, G.: Searching polyhedra by rotating planes, manuscript, arXiv:1104.4137 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Csaba D. Tóth
    • 1
    • 2
  • Godfried T. Toussaint
    • 3
    • 4
  • Andrew Winslow
    • 2
  1. 1.University of CalgaryCalgaryCanada
  2. 2.Tufts UniversityMedfordUSA
  3. 3.New York University Abu DhabiAbu DhabiUnited Arab Emirates
  4. 4.McGill UniversityMontrealCanada

Personalised recommendations