The diameter-constrained Minimum Spanning Tree (MST) problem is as follows: you have a undirected weighted graph $G = (V,E)$ of different weights where $V$ is the set of vertices and $E$ is the set of edges between vertices, and a constant $d$. The goal is to find an MST such that the diameter (i.e., the maximum distance of the shortest paths between vertices) of the MST is at most $d$. My question is that I am debating whether the following is true or not:
Once you have an MST of a graph $G$, if the diameter of the MST is $>d$, then is it true to say that there exist no feasible solution.
Also, once you have a MST of $G$, then is the diameter of that MST the maximum diameter of $G$? or minimum? or what could be said about the diameter of a MST?