Mobile vs. point guards

We study the problem of guarding orthogonal art galleries with horizontal mobile guards (alternatively, vertical) and point guards, using "rectangular vision". We prove a sharp bound on the minimum number of point guards required to cover the gallery in terms of the minimum number of vertical mobile guards and the minimum number of horizontal mobile guards required to cover the gallery. Furthermore, we show that the latter two numbers can be calculated in linear time.Comment: This version covers a previously missing case in both Phase 2 &

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

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.Comment: 20 pages, 12 figure

Covering simple orthogonal polygons with rr-stars

We solve the $r$-star covering problem in simple orthogonal polygons, also known as the point guard problem in simple orthogonal polygons with rectangular vision, in quadratic time.Comment: 22 page

Computing the Difficulty of Critical Bootstrap Percolation Models is NP-hard

Bootstrap percolation is a class of cellular automata with random initial state. Two-dimensional bootstrap percolation models have three universality classes, the most studied being the `critical' one. For this class the scaling of the quantity of greatest interest -- the critical probability -- was determined by Bollobás, Duminil-Copin, Morris and Smith in terms of a combinatorial quantity called `difficulty', so the subject seemed closed up to finding sharper results. In this paper we prove that computing the difficulty of a critical model is NP-hard and exhibit an algorithm to determine it, in contrast with the upcoming result of Balister, Bollobás, Morris and Smith on undecidability in higher dimensions. The proof of NP-hardness is achieved by a reduction to the Set Cover problem

Rooted NNI moves on tree-based phylogenetic networks

We show that the space of rooted tree-based phylogenetic networks is connected under rooted nearest-neighbour interchange (rNNI) moves.Comment: Fixed typos and references to labels in the last subsectio