minimun-spanning-trees

Minimum spanning tree correctness is established by producing safe-edge certificates and constructive proofs using cut, component, and cycle arguments. You verify cut and cycle certificates to confirm that every selected edge belongs to an MST.

Kruskal's algorithm builds a minimum spanning tree by sorting edges and using union-find for connectivity, while Prim's algorithm grows the MST from a single vertex. Both rely on light-edge and cut properties for correctness.

Dynamic MST updates and second-best MST computation require analyzing bottleneck edges and maintaining forest components. You apply cycle and cut arguments to determine which edges can be swapped without violating minimum spanning tree constraints.

Union-find supports Boruvka-style contractions by maintaining forest components and enabling fast connectivity checks during MST construction. It ensures correct handling of component merging across multiple contraction phases.

MST algorithms fail on multi-edges and ties when graph modeling does not ensure proper connectivity and edge weight ordering. You must verify safe-edge certificates and apply cut properties to handle equal-weight edges correctly.

Prim's algorithm is less suitable for minimum spanning tree problems requiring dynamic updates or bottleneck analysis, where Kruskal with union-find or Boruvka-style contractions provide better component tracking and update flexibility.