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Some Results on Hidden Edge Guards

Some Results on Hidden Edge Guards. Sarah Cannon Diane Souvaine Andrew Winslow. Klee’s Art Gallery Problem. Consider the floor plan of an art gallery, and point guards that stand stationary and look in all directions. Victor Klee (1973): How many guards

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Some Results on Hidden Edge Guards

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  1. Some Results on Hidden Edge Guards Sarah Cannon Diane Souvaine Andrew Winslow

  2. Klee’s Art Gallery Problem Consider the floor plan of an art gallery, and point guards that stand stationary and look in all directions. Victor Klee (1973): How many guards are needed to see the entire floor plan?

  3. Hidden Guards In 1989, Thomas Shermer considered the same problem with the restriction that guards cannot see each other. BAD OK

  4. Hidden Guards For hidden guards restricted to vertices, Shermer found that some polygons do not admit guarding! T. Shermer, Hiding people in polygons, Computing, 1989.

  5. Are there other guard varieties for which some polygons do not admit guarding?

  6. Edge Guards

  7. Edge Definitions Historically, edge guards have included the endpoints. Recently, excluding the endpoints has been considered. OPEN CLOSED

  8. Hidden Closed Edge Guards YES NO YES NO

  9. Hidden Open Edge Guards YES NO NO YES

  10. Results for Hidden Edge Guards Open Edges Closed Edges Not all simple polygons admit guarding. Not all monotone polygons admit guarding. Not all starshaped polygons admit guarding. All orthogonal polygons admit guards. Not all orthogonal polygons admit guarding. Guardable polygons may require n-22 edges. Guardable polygons may require (n-12)/3 guards.

  11. Admissibility This polygon cannot be guarded by open or closed hidden edge guards.

  12. Admissibility This polygon cannot be guarded by open or closed hidden edge guards.

  13. Admissibility This polygon cannot be guarded by open or closed hidden edge guards.

  14. Combinatorics Antiforcing gadget

  15. Thank you

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