It's not hard to find examples of graphs mapped to the unit 2-sphere in a way that is unshrinkable, in the sense that there is no homotopy that decreases at least one edge-length without increasing any of the others. For instance, a tetrahedral graph will work. You can think of these physically as net bags that tightly hold a sphere. Puzzle: show that for any such example, there is a two-fold covering space of the graph such that the composed map to the 2-sphere **can** be shrunk, at least slightly. Extra: show that the same holds for graphs mapped to spheres of arbitrary dimension. In at least some cases---for instance, the rounded-out edges of a tetrahedron, octahedron, cube --- this 2-fold covering of the graph can be chosen so the map shrinks all the way to a constant map. The sphere can be extracted from the double cover of the net bag (physically as well as mathematically). I haven't tried constructing these, but they ought to make nice parlor tricks except for the small detail that nobody has a parlor any more. I strongly suspect that for any graph mapped to S^2, some finite sheeted cover can be shrunk all the way to a point. If so, k degree covers are needed, and whether it can always be done in a physically realizable way (i.e. so the strings don't get entangled with each other and you can physically remove the ball from the bag.) Bill Thurston