Abstract
Consider an art gallery formed by a polygon onn vertices withm pairs of vertices joined by interior diagonals, the interior walls. Each interior wall has an arbitrarily placed, arbitrarily small doorway. It is shown in [5] that the minimum number of guards that suffice to guard all art galleries withn vertices andm interior walls is min⌊(2n − 3)/3], ⌊(2m +n)/3⌋, ⌊(2n +m − 2)/4⌋. The proofs for the first two bounds lead to linear-time guard placement algorithms, while this is not known for the third case. We present an alternative short proof for the third upper bound ⌊(2n +m − 2)/4⌋ that also leads to a linear-time guard placement algorithm.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Chazelle, Triangulating a simple polygon in linear time,Discrete Comput. Geom. 6 (1991), 485–524.
V. Chvátal, A combinatorial theorem in plane geometry,J. Combin. Theory Ser. B 18 (1975), 39–41.
S. Fisk, A short proof of Chvátal’s watchman theorem,J. Combin. Theory Ser. B 24 (1978), 374.
J. Hutchinson, Art galleries with walls, Problem #10478,Amer. Math. Monthly 102 (1995), 746.
A. Kündgen, Art galleries with interior walls,Discrete Comput. Geom. 22 (1999) 248–258.
J. O’Rourke,Art Gallery Theorems and Algorithms, The International Series of Monographs on Computer Science, The Clarendon Press, Oxford University Press, New York, 1987.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author thanks The Fields Institute for Research in Mathematical Sciences and the Natural Sciences and Engineering Research Council of Canada for financial support.
Rights and permissions
About this article
Cite this article
Hliněný, P. An addition to art galleries with interior walls. Discrete Comput Geom 25, 311–314 (2001). https://doi.org/10.1007/s004540010078
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s004540010078