Abstract
The art gallery theorem asserts that any polygon with n vertices can be protected by at most ⌊n/3⌋ stationary guards. The original proof by Chvátal uses a nonroutine and nonintuitive induction. We give a simple inductive proof of a new, more general result, the constrained art gallery theorem: If V* and E* are specified sets of vertices and edges that must contain guards, then the polygon can be protected by at most ⌊(n + 2|V*| + |E*|)/3⌋ guards. Our result reduces to Chvátal's art gallery theorem when V* and E* are empty. We give a second short proof of this generalization in the spirit of Fisk's proof of the art gallery theorem using graph colorings.
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