Abstract. We consider beacons, an analog of geographical greedy routing, motivated by sensor network applications. A beacon b is a point that can be activated to create a 'magnetic pull' towards itself everywhere in a polygonal domain P . We explore the properties of beacons and their effect on points in polygons, as well as demonstrate polynomial-time algorithms to compute a variety of structures defined by the action of beacons on P . We establish a polynomial-time algorithm for routing from a point s to a point t using a discrete set of candidate beacons, as well as a 2-approximation and a PTAS for routing between beacons placed without restriction in P .

We explore the art gallery problem for the special case that the domain (gallery) P is an mpolyomino, a polyform whose cells are m unit squares. We study the combinatorics of guarding polyominoes in terms of the parameter m, in contrast with the traditional parameter n, the number of vertices of P . In particular, we show that ⌊ m+1 3 ⌋ point guards are always sufficient and sometimes necessary to cover an m-polyomino, possibly with holes. When m ≤ 3n 4 − 4, the sufficiency condition yields a strictly lower guard number than ⌊ n 4 ⌋, given by the art gallery theorem for orthogonal polygons.

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We consider a generalization of the classical Art Gallery Problem, where instead of a light source, the guards, called k-transmitters, model a wireless device with a signal that can pass through at most k walls. We show it is NP-hard to compute a minimum cover of point 2transmitters, point k-transmitters, and edge 2-transmitters in a simple polygon. The point 2transmitter result extends to orthogonal polygons. In addition, we give necessity and sufficiency results for the number of edge 2-transmitters in general, monotone, orthogonal monotone, and orthogonal polygons. * Abstracts of part of this work appeared in the informal workshops FWCG [7] and EuroCG [8] †