Characterization of greedy TSPs?

Question

Define a greedy tour of a set $S=\{p_1,\ldots,p_n\}$ of $n$ points in $\mathbb{R}^2$ as produced by selecting the $i$-th point $p_i$ to start, and then connecting to the nearest neighbor $p_j$ to $p_i$, then to the nearest neighbor to $p_j$ (excluding $p_i$), and so on, finally closing back to the start $p_i$. So there are $n$ possible greedy tours. Assume general position so there are no ties.

My question is:

When is the shortest greedy tour equal to the Euclidean TSP? Is there any characterization, or at least sufficient conditions on $S$, such that the shortest greedy tour is the TSP?

For example, below shows the optimal TSP for a $50$-point set, and one of the $50$ greedy tours that fails to achieve the TSP.

   
   ^{Optimal tour $62.5$ compared to $77.4$, greedy starting at $(8.6)$.}

Whereas every greedy tour of this set of points on an ellipse is also the TSP tour.

   
   ^$|S|=18$.

Perhaps the "local feature size" or the "reach of a manifold" can be used to quantify conditions on $S$?

Answer 1

A simple "a posteriori" criterion is that on the optimal tour the distances to the tour-neighbors is smaller than that to any of the other vertices.
Convexity alone doesn't suffice as the example of ellipses with sufficiently high excentricity demonstrates.

Another, "a priory" criterion may be that the maximum weight matching of every $K_4$ induced by 4 vertices of the TSP instance consists of the two longest edges of that subgraph.

Ruminating further, a general sufficient condition is that the Minimum Spanning Tree is linear and the edge joining the leaf nodes doesn't cross a tree edge.

The simplest sufficient and generally applicable criterion is however that the set of edges that is the union of the two shortest edges, that are adjacent to a vertex, constitutes to a tour.

Edit
there are counter examples to the criteria tha I have deleted: the instance with points $(0,0),\,(0,1),\,(1,1),\,(1,0),(1/k,1),\,\cdots,\,((k-1)/k,1)$