Topological Art in Simple Galleries

TL;DR

This paper explores the topological structure of minimal guard placements in simple polygons, showing they can represent any semi-algebraic set and characterizing their homeomorphic types.

Contribution

It demonstrates that the space of minimal guard placements can be as complex as any semi-algebraic set and provides explicit instances for various topological spaces.

Findings

V(P) can be homotopy equivalent to any semi-algebraic set.

Instances of the art gallery problem can have guard placement spaces homeomorphic to specific topological spaces.

The Hausdorff distance induces a rich topological structure on guard placement sets.

Abstract

Let $P$ be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in $P$. We say two points $a,b\in P$ can see each other if the line segment $seg(a,b)$ is contained in $P$. We denote by $V(P)$ the family of all minimum guard placements. The Hausdorff distance makes $V(P)$ a metric space and thus a topological space. We show homotopy-universality, that is for every semi-algebraic set $S$ there is a polygon $P$ such that $V(P)$ is homotopy equivalent to $S$. Furthermore, for various concrete topological spaces $T$, we describe instances $I$ of the art gallery problem such that $V(I)$ is homeomorphic to $T$.