In order to efficiently handle cycle detection, we consider each node as part of a tree. When adding an edge, we check if its two component nodes are part of distinct trees. Initially, each node makes up a one-node tree.

```
algorithm kruskalMST'(G: a graph)
sort G's edges by their value
MST = a forest of trees, initially each tree is a node in the graph
for each edge e in G:
if the root of the tree that e.first belongs to is not the same
as the root of the tree that e.second belongs to:
connect one of the roots to the other, thus merging two trees
return MST, which now a single-tree forest
```