We extend a classical Minimum Spanning Tree of a cloud to the new fundamental concept of a Homologically Persistent Skeleton, which is scale-and-rotation invariant and depends only on the given cloud without extra parameters. This graph
(1) is computable in time O(n log n) for any n points in the plane;
(2) has the minimum total length among all graphs that span a 2D cloud at any scale and also have most persistent 1-dimensional cycles;
(3) is geometrically stable for noisy samples around planar graphs.
The preprint is at http://kurlin.org/projects/hopes.pdf