TL;DR
This paper explores extensions of the polygonal art gallery problem, providing new lower bounds for guard placement scenarios and presenting a 3D example with minimal guarding properties.
Contribution
It establishes new minimal polygon sizes for guarding problems and introduces a 3D polyhedron with minimal guard placement characteristics.
Findings
A polygonal fortress with exterior unguardable by guards on every second vertex has at least 12 sides.
An example of a 3D polyhedron with minimal guard coverage properties is provided.
New lower bounds for guard placement in polygonal and polyhedral scenarios are demonstrated.
Abstract
We will consider some extensions of the polygonal art gallery problem. In a recent paper Morrison has shown the smallest (9 sides) example of an art gallery that cannot be observed by guards placed in every third corner. Author also mentioned two related problems, for which the minimal examples are not known. We will show that a polygonal fortress such that its exterior cannot be guarded by sentries placed in every second vertex has at least 12 sides. Also, we will show an example of three-dimensional polyhedron such that its inside cannot be covered by placing guard in every vertex which has both fewer vertices and faces than previously known.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Artificial Intelligence in Games · Optimization and Search Problems