Art gallery theorems

A collection of results similar to Chvátal's original theorem (cf. Chvátal theorem). Each specifies tight bounds on the number of guards of various types that can visually cover a polygonal region. A guard sees a point if and only if there is an such that the segment is nowhere exterior to the closed polygon . A polygon is covered by a set of guards if every point in the polygon is visible to some guard. Types of guards considered include point guards (any point ), vertex guards ( is a vertex of ), diagonal guards ( is a nowhere exterior segment between vertices), and edge guards ( is an edge of ). An art gallery theorem establishes the exact bound for specific types of guards in a specific class of polygons, where is the maximum over all polygons with vertices, of the minimum number of guards that suffice to cover .

For simple polygons, the main bounds for are: vertex guards (the Chvátal theorem), diagonal guards [a6], and vertex guards for orthogonal polygons (polygons whose edges meet orthogonally) [a5]. No tight bound for edge guards has been established.

Attention can be turned to visibility outside the polygon: for coverage of the exterior of the polygon, one needs point guards ( any point not in the interior of ) [a6]; for coverage of both interior and exterior, one needs vertex guards [a2]. For polygons with holes and a total of vertices, the main results are: point guards for simple polygons [a1], [a4], and point guards for orthogonal polygons (independent of ) [a3]. Both hole problems remain open for vertex guards. See [a7] for a survey updating [a6].

References

[a1]	I. Bjorling-Sachs, D. Souvaine, "An efficient algorithm for guard placement in polygons with holes" Discrete Comput. Geom. , 13 (1995) pp. 77–109
[a2]	Z. Füredi, D. Kleitman, "The prison yard problem" Combinatorica , 14 (1994) pp. 287–300
[a3]	F. Hoffmann, "On the rectilinear art gallery problem" , Proc. Internat. Colloq. on Automata, Languages, and Programming 90 , Lecture Notes in Computer Science , 443 , Springer (1990) pp. 717–728
[a4]	F. Hoffmann, M. Kaufmann, K. Kriegel, "The art gallery theorem for polygons with holes" , Proc. 32nd Found. Comput. Sci. (1991) pp. 39–48
[a5]	J. Kahn, M. Klawe, D. Kleitman, "Traditional galleries require fewer watchmen" SIAM J. Algebraic Discrete Methods , 4 (1983) pp. 194–206
[a6]	J. O'Rourke, "Art gallery theorems and algorithms" , Oxford Univ. Press (1987)
[a7]	T. Shermer, "Recent results in art galleries" Proc. IEEE , 80 : 9 (1992) pp. 1384–1399