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The statement of the problem is incorrect. But $T$-joins are indeed very much related to the perfect matching problem. What the theorem that 9.3a is supposed to be conveying is: Assume $G$ is connected. Suppose that $T = V$. The minimum $T$-join can be found as follows: construct a complete graph $G'$ such that the weight on an edge (a,b) in $G'$ is the ...


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