Mapping a Polygon with Holes Using a Compass

We consider a simple robot inside a polygon \(\mathcal{P}\) with holes. The robot can move between vertices of \(\mathcal{P}\) along lines of sight. When sitting at a vertex, the robot observes the vertices visible from its current location, and it can use a compass to measure the angle of the boundary of \(\mathcal{P}\) towards north. The robot initially only knows an upper bound \(\bar{n}\) on the total number of vertices of \(\mathcal{P}\). We study the mapping problem in which the robot needs to infer the visibility graph G vis of \(\mathcal{P}\) and needs to localize itself within G vis. We show that the robot can always solve this mapping problem. To do this, we show that the minimum base graph of G vis is identical to G vis itself. This proves that the robot can solve the mapping problem, since knowing an upper bound on the number of vertices was previously shown to suffice for computing G vis.