Undirected Eulerian check
All degrees even → Eulerian circuit is the answer. Otherwise: extra edges needed.
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Odd vertex matching
Find shortest paths between all pairs of odd-degree vertices. Add duplicate edges (min matching) so degrees even. Then find Eulerian.
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Directed variant
NP-hard in general. Special case (mixed graphs) polynomial via flow.
Complexity
Undirected: O(V³) via Floyd + matching. Practical for moderate V.