Finding a Minimal Tree in a Polygon with its Medial Axis

In order to solve a problem arising when generalizing topographical maps, we consider the following problem for simple polygons, i.e., coherent polygons without holes. Some edges of the polygon may be marked as hard, and at least two vertices of the polygon are marked as terminals. We show that the problem to find a tree of minimum total length, spanning the hard edges and terminals, using only edges of the polygon and its medial axis, can be stated as the problem to find a minimum Steiner tree in a Halin graph, and can be efficiently solved in linear time