Kruskal's algorithm sorts the edges and accepts an edge precisely when it joins two currently different connected components of a graph. A disjoint-set data structure maintains those components:
Kruskal(V, E, w):
    A = empty set
    make one disjoint set for each vertex in V
    for (u,v) in E sorted by nondecreasing w:
        if find(u) != find(v):
            add (u,v) to A
            union(u,v)
            if size(A) == size(V) - 1:
                break
    return A
An accepted edge merges two trees of the current forest. An edge within one component would complete a graph cycle and is skipped. On connected input, after acceptances the forest is a spanning tree. For a one-vertex graph the empty set is already the answer.

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