Special Guards in Chromatic Art Gallery

TL;DR

This paper introduces new variants of the chromatic art gallery problem using restricted-angle and orthogonal guards, demonstrating constant and logarithmic bounds on the number of colors needed for effective guarding.

Contribution

It presents novel bounds for the chromatic guarding number with restricted-angle and orthogonal guards, including constant bounds for special polygons and logarithmic bounds for general orthogonal polygons.

Findings

Strong chromatic guarding number is constant for 0-180 degree guards.

Orthogonal guards require logarithmic number of colors for general polygons.

Special orthogonal polygons like snake, staircase, and convex polygons need constant colors.

Abstract

We present two new versions of the chromatic art gallery problem that can improve upper bound of the required colors pretty well. In our version, we employ restricted angle guards so that these modern guards can visit $\alpha$-degree of their surroundings. If $\alpha$ is between $0$ and $180$ degree, we demonstrate that the strong chromatic guarding number is constant. Then we use orthogonal $90$-degree guards for guarding the polygons. We prove that the strong chromatic guarding number with orthogonal guards is the logarithmic order. First, we show that for the special cases of the orthogonal polygon such as snake polygon, staircase polygon and convex polygon, the number of colors is constant. We decompose the polygon into parts so that the number of the conflicted parts is logarithmic and every part is snake. Next, we explain the chromatic art gallery for the orthogonal polygon with guards that have rectangular visibility. We prove that the strong chromatic guarding number for orthogonal polygon with rectangular guards is logarithmic order, too. We use a partitioning for orthogonal polygon such that every part is a mount, then we show that a tight bound for strong chromatic number with rectangular guards is $\theta(\log n)$