On Some City Guarding Problems

We consider guarding a city of kvertical buildings, each having a rectangular base, by placing guards only at vertices. The aim is to use the smallest number of guards. The problem is a 2.5D variant of the traditional art gallery problem, and finds applications in urban security. We give upper and lower bounds on the number of guards needed for a few versions of the problem. Specifically, we prove that $\lfloor\frac{2(k-1)}{3}\rfloor + 1$ guards are always sufficient and sometimes necessary to guard all roofs, and $1 + k + \lfloor \frac{k}{2}\rfloor$ guards are always sufficient to guard the roofs, walls, and the ground, while each roof has at least one guard on it.