Guarding curvilinear art galleries with edge or mobile guards via 2-dominance of triangulation graphs

Abstract

In 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) ⌊ n + 1 3 ⌋ diagonal guards are always sufficient and sometimes necessary, and (2) ⌊ 2 n + 1 5 ⌋ 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 ⌊ n + 1 3 ⌋ in linear time and space, (2) an edge 2-dominating set of size ⌊ 2 n + 1 5 ⌋ in O ( n 2 ) time and O ( n ) space, and (3) an edge 2-dominating set of size ⌊ 3 n 7 ⌋ 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 ⌊ n + 1 3 ⌋ in O ( n log n ) time, (2) an edge guard set of size ⌊ 2 n + 1 5 ⌋ in O ( n 2 ) time, and (3) an edge guard set of size ⌊ 3 n 7 ⌋ in O ( n log n ) time. All space requirements are linear. Finally, we show that ⌊ n 3 ⌋ mobile or ⌈ n 3 ⌉ edge guards are sometimes necessary. When restricting our attention to monotone piecewise-convex polygons, the bounds mentioned above drop: ⌈ n + 1 4 ⌉ edge or mobile guards are always sufficient and sometimes necessary; such an edge or mobile guard set, of size at most ⌈ n + 1 4 ⌉ , can be computed in O ( n ) time and space.