 Solution

It is single-linkage clustering. Its intercluster distance is the smallest cross-pair Euclidean distance. Successive nearby points can join around the ring and within the dense central cloud before any edge crosses the gap between them. Equivalently, at a threshold below the between-group gap but above the lengths needed to connect each group, the single-linkage clustering components are precisely the central group and the ring.

Complete-linkage clustering uses the largest cross-pair Euclidean distance. Opposite points on the ring are far apart, so an intact ring has a large diameter. That rule favors compact pieces and can attach a ring segment to the center before all ring segments join. The visible gap and short links within each group explain why single-linkage clustering recovers the two intended clusters in cluster analysis in this example; it is not a universal guarantee for arbitrary noisy rings.