Orthogonal Art Galleries with Holes: A Coloring Proof of Aggarwal's Theorem

We prove that n+h 4 vertex guards are always sufficient to see the entire interior of an n-vertex orthogonal polygon P with an arbitrary number h of holes provided that there exists a quadrilateralization whose dual graph is a cactus. Our proof is based upon 4-coloring of a quadrilateralization graph, and it is similar to that of Kahn and others for orthogonal polygons without holes. Consequently, we provide an alternate proof of Aggarwal's theorem asserting that n+h 4 vertex guards always suffice to cover any n-vertex orthogonal polygon with h 2 holes.

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: Pawel Zylinski
• Keywords: COMBINATORICS 2006, N-vertex Orthogonal Polygon, Orthogonal Polygon, Vertex Guards