M-Guarding in K-Visibility

TL;DR

This paper investigates M-guarding polygons with holes using k-visibility guards, establishing new theoretical bounds and providing algorithms for guard placement to ensure multiple coverage levels.

Contribution

It introduces a theorem for 2-guarding polygons with holes under k-visibility and presents an algorithm based on convex decomposition for M-guarding.

Findings

Any polygon with holes can be 2-guarded under k-visibility for k ≥ 2.

Every point can be visible to at least four 2-visibility guards.

For even k ≥ 2, points can be visible to k + 2 guards.

Abstract

We explore the problem of $M$-guarding polygons with holes using $k$-visibility guards, where a set of guards is said to $M$-guard a polygon if every point in the polygon is visible to at least $M$ guards, with the constraint that there may only be 1 guard on each edge. A $k$-visibility guard can see through up to $k$ walls, with $k \geq 2$. We present a theorem establishing that any polygon with holes can be 2-guarded under $k$-visibility where $k \geq 2$, which expands existing results in 0-visibility. We provide an algorithm that $M$-guards a polygon using a convex decomposition of the polygon. We show that every point in the polygon is visible to at least four $2$-visibility guards and then extend the result to show that for any even $k \geq 2$ there exists a placement of guards such that every point in the polygon is visible to $k + 2$ guards.