Kruskal's algorithm

Kruskal's algorithm finds a safe edge to add to the growing forest by finding, of all the edges that connect any two trees in the forest, an edge $(u,v)$ with the lowest weight. Let

$C_1$ and $C_2$ denote the two trees that are connected by $(u,v).$ Since $(u,v)$ must be a light edge connecting $C_1$ to some other tree, Corollary $21\mathrm{.}2$ implies that $(u,v)$ is a safe edge

for $C_1.$ Kruskal's algorithm qualifies as a greedy algorithm because at each step it adds to the forest an edge with the lowest possible weight.