Guarding curvilinear art galleries with edge or mobile guards

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. 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. We also show that a diagonal (resp., edge) 2-dominating set of size ⌊2n+1/5⌋ can be computed in O(n2) time and O(n) space. Based on these results we prove that, in order to monitor a piecewise-convex polygon P witn n ≥ 2 vertices: (1) ⌊n+1/3⌋ mobile guards or ⌊2n+1/5⌋ edge guards are always suffcient, and (2) ⌊n/3⌋ mobile edge guards are sometimes necessary. A mobile (resp., edge) guard set for P of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ can be computed in O(n log n + T(n)) time and O(n) space, where T(n) denotes the time for computing a diagonal (resp., edge) 2-dominating set of size ⌊n+1/3⌋ (resp., ⌊2n+1/5⌋ or ⌊3n/7⌋ for a trinagulation graph with n vertices.