Characterizing and recognizing LR-visibility polygons

Authors:
Xuehou Tan;Bo Jiang;Jing Zhang
Affiliations:
School of Information Science and Technology, Dalian Maritime University, Linghai Road 1, Dalian, China and School of Information Science and Technology, Tokai University, 4-1-1 Kitakaname, Hirats ...;School of Information Science and Technology, Dalian Maritime University, Linghai Road 1, Dalian, China;School of Information Science and Technology, Dalian Maritime University, Linghai Road 1, Dalian, China
Venue:
Discrete Applied Mathematics
Year:
2014

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Abstract

A simple polygon P is LR-visible if there are two points s, t on the boundary of P such that every point on the clockwise boundary of P from s to t is visible from some point of the other boundary of P from t to s and vice versa. We show that P is not LR-visible if and only if it has k non-redundant components such that each of them exactly intersects with k^' other components, where 0@?k^'@?k-3. Our characterization is obtained by investigating the structure of the considered non-redundant components and representing it by a set of directed chords of a circle. Furthermore, we develop a simple O(n) time algorithm for determining whether a given polygon with n vertices is LR-visible as well as for reporting a pair or all pairs (s,t) which admit LR-visibility. This greatly simplifies the existing algorithm for recognizing LR-visibility polygons. Also, our result can be used to simplify the existing solutions of other LR-visibility problems.