Guarding the walls of an art gallery

P so that each point of P is seen by at least one guard. We introduce and explore the edge-covering problem; the guards are required to observe the edges of P; metaphorically the paintings on the walls of the art gallery, and not necessarily every interior point. We compare minimum edge and interior covers for a given polygon and analyze the bounds and complexity for the edge-covering problem. We also introduce and analyze a restricted edge covering problem, where full visibility of each edge from at least one guard is required. For this problem we present an algorithm that computes a set of regions where a minimum set of guards must be located. The algorithm can also deal with the external visibility of a set of polygons.