Is this homework problem on T-joins wrong? In Question 9.3a, it states that if $T=V$, then the minimum cost perfect matching is the minimum cost T-join. Is this actually true? I think I have a counterexample which I have drawn below.
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 length of the shortest path between vertices $a$ and $b$ in $G$. Now, find a minimum weight perfect matching in $G'$. This gives the minimum $T$-join in $G$.