Spanning trees and especially minimum spanning trees are the anchor point of a whole class of TSP heuristics, most prominently the Christofides algorithm.
I noticed however that there may be MSTs for which the generated tours may not even be two-optimal, namely if some or all of its leaf nodes would be encircled by tree edges:
the Christofides algorithm would connect both leaf nodes with an edge, generating a tour with arbitrarily many self-crossings.
Interestingly that spiraling in MSTs apparently isn’t honored in benchmark instances for TSP heuristics.
is it possible to efficiently calculate vertex weights that, when added to the weights of adjacent vertices, guarantee that the tours generated from the MST will be two-optimal?