In graph theory, a spanning tree of a connected graph is a subgraph that is both a tree (connected and acyclic) and includes every vertex of the original graph; every connected graph provably has at least one spanning tree, established via a minimality/contradiction argument showing that any minimum-edge connected subgraph spanning all vertices cannot contain a cycle. For edge-weighted connected graphs, a minimum spanning tree—the spanning tree minimizing the sum of edge weights—can be constructed by a greedy algorithm that repeatedly adds the lowest-weight edge not creating a cycle, with correctness established via an inductive exchange argument showing the greedily selected edge set can always be extended to a minimum spanning tree.
In graph theory, a spanning tree of a connected graph is a subgraph that is both a tree (connected and acyclic) and includes every vertex of the original graph; every connected graph provably has at …