A tree in a undirectional graph has |V|-1
edges.
Since a tree is the connected component with least edges as possible - it basically means that for each connected undirectional graph, |E| is in Omega(|V|)
, so |V| is dominated by |E|.
This basically means that if |E| < |V|-1
- the graph is not connected.
Now, since Kruskal algorithm is designed to find a spanning tree, you can abort the algorithm once you have found |E| < |V|-1
- there is no spanning tree at all, no point to look for one.
From this we conclude that when |E| < |V|-1
, there is no point in discussing complexity of Kruskal Algorithm, and we can safely assume that |E| >= |V| -1
, so |V|
is dominated by |E|
.