Reconstructing Visibility Graphs with Simple Robots

We consider the problem of finding a minimalistic configuration of sensors that enable a simple robot inside an initially unknown polygon \mathcal{p}\mathcal{p} on n vertices to reconstruct the visibility graph of \mathcal{p}\mathcal{p}. The robot can sense features of its environment through its sensors, and it is allowed to move from vertex to vertex.we aim at understanding which sensorial capabilities are sufficient for the reconstruction of the visibility graph of \mathcal{p}\mathcal{p}. We are able to show that the combinatorial visibilities at every vertex do not contain enough information even when combined with the knowledge of the exact interior angle at each vertex. Using sensors that can put distant vertices into a spatial relation on the other hand can in some cases enable our robot to reconstruct the visibility graph of \mathcal{p}\mathcal{p}. We show that this is true for a sensor that can distinguish whether the angle between two vertices the robot sees is convex or reflex, as long as the robot is capable of identifying the vertex it last visited. We also show that measuring angles exactly is enough, if the robot has a compass.