Grid-Obstacle Representations with Connections to Staircase Guarding

In this paper, we study grid-obstacle representations of graphs where we assign grid-points to vertices and define obstacles such that an edge exists if and only if an $xy$-monotone grid path connects the two endpoints without hitting an obstacle or another vertex. It was previously argued that all planar graphs have a grid-obstacle representation in 2D, and all graphs have a grid-obstacle representation in 3D. In this paper, we show that such constructions are possible with significantly smaller grid-size than previously achieved. Then we study the variant where vertices are not blocking, and show that then grid-obstacle representations exist for bipartite graphs. The latter has applications in so-called staircase guarding of orthogonal polygons; using our grid-obstacle representations, we show that staircase guarding is \textsc{NP}-hard in 2D.Comment: To appear in the proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

On rr-Guarding Thin Orthogonal Polygons

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of $r$-guards is polynomial for tree polygons, but the run-time was $\tilde{O}(n^{17})$. We show here that with a different approach the running time becomes linear, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in $h+K$.Comment: 18 page

Securing Pathways with Orthogonal Robots

The protection of pathways holds immense significance across various domains, including urban planning, transportation, surveillance, and security. This article introduces a groundbreaking approach to safeguarding pathways by employing orthogonal robots. The study specifically addresses the challenge of efficiently guarding orthogonal areas with the minimum number of orthogonal robots. The primary focus is on orthogonal pathways, characterized by a path-like dual graph of vertical decomposition. It is demonstrated that determining the minimum number of orthogonal robots for pathways can be achieved in linear time. However, it is essential to note that the general problem of finding the minimum number of robots for simple polygons with general visibility, even in the orthogonal case, is known to be NP-hard. Emphasis is placed on the flexibility of placing robots anywhere within the polygon, whether on the boundary or in the interior.Comment: 8 pages, 5 figure