Guarding Disjoint Orthogonal Polygons in the Plane

Abstract

Given a set F of disjoint monotone orthogonal polygons, we present bounds on the number of vertex guards required to monitor the free space and the boundaries of the polygons in F. We assume that the range of vision of a guard is bounded by 180\(^{\circ }\) (the region in front of the guard). For k disjoint axis-aligned monotone orthogonal polygons \(H_1, H_2, \dots , H_k\) with \(m_1, m_2, \dots , m_k\) vertices respectively, such that \(\sum _{i = 1}^{k} m_i = m\), we prove that \(\frac{m}{2} + \lfloor \frac{k}{4} \rfloor + 4\) vertex guards are sufficient to monitor the boundaries of the polygons and the free space. When the orthogonal polygons are arbitrary oriented, we show that \(\frac{m}{2} + k + 1\) vertex guards are sometimes necessary to monitor the boundaries and the free space and conjecture the bound is tight.