M-Guarding in K-Visibility

TL;DR

This paper extends the Art Gallery Problem to $M$-guarding under $k$-visibility, showing that polygons with holes can be 2-guarded when $k \ge 2$ and providing a convex-decomposition based algorithm to place edge guards to achieve $M$-visibility. It proves that every point is visible to at least four $2$-visibility guards and, for any even $k \ge 2$, constructs guard placements so that every point is visible to at least $k+2$ guards; it also offers a general bound of $kC$ guards for a polygon decomposed into $C$ convex pieces. The method relies on optimal convex decomposition, dual graphs, and careful guard repositioning to maintain $k$-visibility as pieces are merged, with a focus on redundancy over minimization. The results have potential applications in wireless mapping and coverage problems where multiple lines of sight and edge-restricted placements are important, providing a constructive framework for guaranteed multi-coverage under $k$-visibility.

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.

Figures (4)

• Figure 1: The ray is cast from $v_1$ along the reflex chain that forms the pocket. Once we find the ($k/2$)-th critical reflex vertex (orange ray), the sweep ends
• Figure 2: An invalid polygon with a hole. The vertex $H$ intersects the boundary of the outer polygon, violating the requirement that holes must lie entirely within the outer polygon
• Figure 3: A polygon that demonstrates the size of the valid area for placing a guard in the adjacent piece.
• Figure 4: Panels (a) and (b) show different steps of Algorithm \ref{['alg:pseudo_2']}.

Theorems & Definitions (14)

• theorem thmcountertheorem
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof
• lemma thmcounterlemma
• proof : Proof by Contradiction
• lemma thmcounterlemma
• proof