  We apply combinatorial methods to a geometric problem: the classification of
polytopes, in terms of Minkowski decomposability. Various properties of
skeletons of polytopes are exhibited, each sufficient to guarantee
indecomposability of a significant class of polytopes. We illustrate further
the power of these techniques, compared with the traditional method of
examining triangular faces, with several applications. In any dimension $d\neq
2$, we show that of all the polytopes with $d^2+\frac{d}{2}$ or fewer edges,
only one is decomposable. In 3 dimensions, we complete the classification, in
terms of decomposability, of the 260 combinatorial types of polyhedra with 15
or fewer edges.
