  On a convex polyhedron P, the cut locus C(x) with respect to a point x is a
tree of geodesic segments (shortest paths) on P that includes every vertex. We
say that P has a skeletal cut locus if there is some x in P such that C(x) is a
subset of Sk(P), where Sk(P) is the 1-skeleton of P . At a first glance, there
seems to be very little relation between the cut locus and the 1-skeleton, as
the first one is an intrinsic geometry notion, and the second one specifies the
combinatorics of P.
  In this paper we study skeletal cut loci, obtaining four main results. First,
given any combinatorial tree T without degree-2 nodes, there exists a convex
polyhedron P and a point x in P with a cut locus that lies in Sk(P), and whose
combinatorics match T. Second, any (non-degenerate) polyhedron P has at most a
finite number of points x for which C(x) is a subset of Sk(P). Third, we show
that almost all polyhedra have no skeletal cut locus. Fourth, we provide a
combinatorial restriction to the existence of skeletal cut loci.
  Because the source unfolding of P with respect to x is always a
non-overlapping net for P, and because the boundary of the source unfolding is
the (unfolded) cut locus, source unfoldings of polyhedra with skeletal cut loci
are edge-unfoldings, and moreover "blooming," avoiding self-intersection during
an unfolding process.
