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.

翻译：在凸多面体 P 上，关于点 x 的割迹 C(x) 是 P 上包含所有顶点的测地线段（最短路径）树。若存在 P 中的某点 x 使得 C(x) 为 Sk(P) 的子集（其中 Sk(P) 是 P 的一维骨架），则称 P 具有骨架割迹。初看之下，割迹与一维骨架之间似乎关联甚微，因为前者属于内蕴几何概念，而后者则规定了 P 的组合结构。本文研究骨架割迹，获得四项主要结果。其一，对任意不含二度结点的组合树 T，存在凸多面体 P 及其上点 x，使得割迹位于 Sk(P) 中且其组合结构与 T 匹配。其二，任何（非退化）多面体 P 至多存在有限个点 x 使得 C(x) 是 Sk(P) 的子集。其三，我们证明几乎所有多面体均无骨架割迹。其四，我们给出骨架割迹存在性的组合限制条件。由于 P 关于 x 的源展开始终是 P 的无重叠展开网，且源展开的边界即为（展开后的）割迹，因此具有骨架割迹的多面体的源展开可实现边展开，并进一步具备"绽放"特性——在展开过程中避免自相交。