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Boundary-Boundary Guarding ER-Completeness

Determine whether boundary-boundary guarding—where all guards must be placed on the polygon boundary and only the boundary must be guarded—is ER-complete.

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Background

The standard Art Gallery Problem is ER-complete; the boundary guarding variant restricts guards to the polygon boundary while requiring visibility of the interior.

A further variant, boundary-boundary guarding, restricts both guard placement and the guarded region to the boundary; its complexity status is not resolved.

References

It is open if boundary-boundary guarding is -complete; in the boundary-boundary variant, all guards must be placed on the boundary, and only the boundary has to be guarded.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in ProblemCG “Boundary-Guarding” entry