Guarding curvilinear art galleries with edge or mobile guards

In this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves. We consider two types of guards: edge guards (these are edges of the polygon) and mobile guards (these are either edges or straight-line diagonals of the polygon). Our focus is on piecewise-convex polygons, i.e., polygons that are locally convex, except possibly at the vertices, and their edges are convex arcs. We reduce the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a constrained triangulation graph with edges or diagonals, where 2-dominance requires that every triangle in the triangulation graph has at least two of its vertices in the 2-dominating set. We show that, given a triangulation graph TP of a polygon P with n 3 vertices: (1) n+1 3 diagonal guards are always sufficient and sometimes necessary, and (2) 2n+1 5 edge guards are always sufficient and 2n 5 edge guards are sometimes necessary, in order to 2-dominate TP...

• Type: Journal
• Year: 2008
• Where: SMA
• Authors: Menelaos I. Karavelas
• Keywords: Edge, Polygon, SMA 2008, Solid Modeling, Triangulation Graph