Mapping a polygon with holes using a compass

We consider a simple robot inside a polygon p with holes. The robot can move between vertices of 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 p towards north. The robot initially only knows an upper bound n¯ on the total number of vertices of p. We study the mapping problem in which the robot needs to infer the visibility graph gvis of p and needs to localize itself within gvis. We show that the robot can always solve this mapping problem. To do this, we show that the minimum base graph of gvis is identical to gvis 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 gvis.