Somebody proposes the following recursive algorithm to find a minimum

spanning tree $($MST$)$ of a connected undirected graph G $=($V$,$ E$)$ with

edge weights:

$$First$,$ partition the nodes V into two non$-$empty sets, S and V $\backslash$ S$,$ so

that each of the resulting parts of the graph, call them GS and GV $\backslash$S$,$ is

connected. Second, recursively find a MST TS for the subgraph GS$,$ and

a MST TV $\backslash$S for the subgraph GV $\backslash$S$.$ Third, construct from TS and TV $\backslash$S

a spanning tree for G by choosing from all edges $($v$,$ w$)$ in E with v in S

and w in $($V $\backslash$ S$),$ the one of minimum weight.$$

Justify whether this algorithm always finds a MST of G $($for example, by

demonstrating that it resembles Kruskal$$s or Prim$$s algorithms$),$ or give

a counterexample to the proposed algorithm.