This greedy strategy is captured by the following "generic" algorithm, which grows the minimum spanning tree one edge at a time. The algorithm manages a set A that is always a subset of some minimum spanning tree. At each step, an edge (u, v) is determined that can be added to A without violating this invariant, in the sense that A {(u, v)} is also a subset of a minimum spanning tree. We call such an edge a safe edge for A, since it can be safely added to A without destroying the invariant.

 

GENERIC−MST(G, w)
1 A
2 while A does not form a spanning tree
3     do find an edge (u, v) that is safe for A
4         A <- A {(u, v)}
5 return A

 

Note that after line 1, the set A trivially satisfies the invariant that it is a subset of a minimum spanning tree.
The loop in lines 2−4 maintains the invariant. When the set A is returned in line 5, therefore, it must be a minimum spanning tree.
The tricky part is, of course, finding a safe edge in line 3. One must exist, since when line 3 is executed, the invariant dictates that there is a spanning tree T such that A T, and if there is an edge (u,v) T such that (u, v) A, then (u, v) is safe for A.