Partitioning orthogonal polygons into at most 8-vertex pieces, with application to an art gallery theorem (1509.05227v4)

Abstract: We prove that every simply connected orthogonal polygon of $n$ vertices can be partitioned into $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that $\left\lfloor\frac{3 n +4}{16}\right\rfloor$ mobile guards are sufficient to control the interior of an $n$-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only required at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answers to two questions of O'Rourke. Our result is also a further example of the "metatheorem" that (orthogonal) art gallery theorems are based on partition theorems.