Why it matters

General graph matching harder than bipartite. Understanding shapes advanced graph problems.

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The architecture

Augmenting path: alternates matched + unmatched edges.

Blossom: odd cycle. Contract to super-vertex.

Blossom algorithmFind augmentingBFS from unmatchedBlossom contractionhandle odd cyclesAugmentflip matched statusComplex implementation; use library implementations
Blossom algorithm.
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How it works end to end

Blossom found during BFS: contract, search on quotient graph. Expand on augmenting.

Complexity: O(V³) or better with efficient implementations.

Weighted version: max-weight matching, more complex.