Why it matters

König's theorem enables bipartite optimization. Understanding shapes combinatorics.

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The architecture

Compute max matching.

Construct min vertex cover from unmatched vertices.

Sizes equal.

König's theoremMax matchingsize MConstructive coverfrom alternating BFSMin vertex coversize MBipartite: matching + cover polynomial; general graphs NP-hard for cover
König's theorem.
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How it works end to end

Alternating BFS from unmatched vertices.

Cover: matched vertices on visited side + unmatched neighbors.

Certificate of optimality.