The prim algorithm proves:
For the ai points in the minimum cost spanning tree
delete k edges connected to him
The minimal connected graph becomes k+1 connected subgraphs
Select the minimum edge a1 to a2 connecting the ai point to the outside world
must be an edge in the smallest tree
For two points a1 a2 in the minimum cost spanning tree
delete the k` edges they connect to the outside world
The connected graph becomes k`+1 connected subgraphs
The smallest edge ai to a3 where they connect to the outside world
must be an edge in the tree
........
According to the above steps, the n-1 edges of all n points can be determined
kruskal proof
It can be known from the prim algorithm
The minimum edge connecting a subgraph of a minimum spanning tree to the outside world must be an edge in the minimum spanning tree
The first edge selected according to the kruskal algorithm
a1 to a2, is the smallest edge of a1 connecting the outside world
It can be known from the prim algorithm that it belongs to the minimum spanning tree
Second edge a3 (may be equal to a1 or a2) to a4 (may be equal to a1 or a2) found by kruskal's algorithm
Take the minimum spanning tree to which a3 belongs to the minimum edge connected to the outside world (the smaller one has either been selected or has formed a loop with the determined edge)
It can be known from the prim algorithm that it belongs to the minimum spanning tree
........
According to the above steps, the n-1 edges of all n points can be determined