7 research outputs found
Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs
AbstractIn this paper we consider the problem of monitoring an art gallery modeled as a polygon, the edges of which are arcs of curves, with edge or mobile guards. 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 transform the problem of monitoring a piecewise-convex polygon to the problem of 2-dominating a properly defined 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 its 2-dominating set. We show that: (1) ân+13â diagonal guards are always sufficient and sometimes necessary, and (2) â2n+15â edge guards are always sufficient and sometimes necessary, in order to 2-dominate a triangulation graph. Furthermore, we show how to compute: (1) a diagonal 2-dominating set of size ân+13â in linear time and space, (2) an edge 2-dominating set of size â2n+15â in O(n2) time and O(n) space, and (3) an edge 2-dominating set of size â3n7â in O(n) time and space.Based on the above-mentioned results, we prove that, for piecewise-convex polygons, we can compute: (1) a mobile guard set of size ân+13â in O(nlogn) time, (2) an edge guard set of size â2n+15â in O(n2) time, and (3) an edge guard set of size â3n7â in O(nlogn) time. All space requirements are linear. Finally, we show that ân3â mobile or ân3â edge guards are sometimes necessary.When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ân+14â edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ân+14â, can be computed in O(n) time and space