# Is this homework problem on T-joins wrong? [closed]

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 claim is true of the G is a metric space. That is, if you take the shortest path distances induced by the edge lengths and complete the graph then T-join on V is same as min-cost perfect matching. Nov 1 '20 at 15:10

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 $$G$$.