The art gallery problem is formulated in geometry as the minimum number of guards that need to be placed in an n-vertex simple polygon such that all points of the interior are visible. Visibility is defined such that two points u and v are mutually visible if the line segment joining them lies inside the polygon

1. Mathematical Problem Solving
Guards
Olive Byukusenge,Sabah Mirghani Yagoub Mohammed, Akor Stanley
Yusuf Brima, Patrick Ntwari Shema
Group Seven(7)
October 9, 2020
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2. Presentation Outline
Introduction
Examples
Theorems
Proofs
Applications
Conclusion
References
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3. Problem formulation
Consider a plot of farm owned by Brian, what is the minimum
number of stationary guards needed to protect the farm?
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4. Problem formulation
Given a n-vertex simple non-regular polygon, what is the
minimum number of guards to see every point of the interior
of the polygon?
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5. Polygons
Regular polygon
Non-regular polygons
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6. Examples
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7. Theorem
Theorem
Every polygon can be triangulated, it is possible to split a complex
polygon to smaller triangles [1] [2].
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8. Theorem
Number of line segments equal to n − 3 and where n is the number of
sides of the polygon.
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9. Proofs
Proof.
Base Case:
Considering the simplest polygon (a triangle) with sides n = 3, the
triangulation theorem works perfectly.
Next we try to triangulate any n-sided polygon and observe that they can
indeed be triangulated.
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10. Theorem
Theorem
For any triangulated polygon, we can assign three colours to the vertices,
such that all the vertices of that polygon are expressed in three colours.
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11. Proof
Proof.
Base Case:
Considering the very trivial case of a simple polygon [that is a triangle]
where n = 3, we actually see that for each vertex we can assign a
different colour.
The 3-colorable theorem can then be generalized for larger polygons
n = 4, 5, ........., k and this holds true as shown in the figure below.
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12. Theorem
Figure 4: 3-colorable theorem
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13. Theorem
Figure 5: 3-colorable theorem
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14. Theorem
Figure 6: 3-colorable theoremGuard Problem Group Seven(7) October 9, 2020 14 / 18

15. Applications
Sensor Networks
Robotics (motion localization)
e.t.c.
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16. Conclusion
The minimum number of guards required by Brian to secure the entire
perimeter of his farm is 7 as indicated by the red colours of the 24-sided
polygons in figure 6.
The number of guards required to cover any n-gon shaped field is
proportional to the number of sides of the polygon, the solution is found
by taking the floor of n
3 where n is the number of sides of the n-gon,
however this value is not often the minimum number of guards
required.
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17. End
Thank you
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18. References
[1] N. Chesnokov, The art gallery problem: An overview and extension
to chromatic coloring and mobile guards, 2018.
[2] C. D. T´oth, G. T. Toussaint, and A. Winslow, “Open guard edges
and edge guards in simple polygons,” in Spanish Meeting on
Computational Geometry, Springer, 2011, pp. 54–64.
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