Connecting all vertices with a tree
A spanning tree includes every vertex and is connected and acyclic, using V−1 edges for V vertices. The optimization compares sums across complete spanning trees, not whether every selected edge is individually smallest in the whole graph.
Weights can represent abstract costs rather than geometric distance and may be zero or negative. A disconnected graph has no single spanning tree; minimum spanning trees for its components instead form a minimum spanning forest.
Cuts justify safe local choices
A cut partitions vertices into two disjoint sets, with crossing edges joining them. When existing selected edges do not cross it, a lightest crossing edge can safely extend a compatible optimum, supporting Prim and Kruskal greedy methods.
Distinct weights ensure uniqueness. Ties can permit several equally good trees without invalidating the algorithms. A selected tied edge should therefore not be described as the uniquely mandatory edge in every optimum.
Connection cost differs from path length
A shortest-path tree minimizes source-to-vertex path lengths, whereas an MST minimizes the total weight of all tree edges. Sharing inexpensive connections can lower that total while making some routes longer, so the objectives generally yield different trees.
Networks may also require redundancy or connectivity after failure. Removing any tree edge disconnects the tree, so an MST does not automatically satisfy those reliability constraints. Its cost optimum applies only to the stated tree model.
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