This is a homework problem I've been given and I've been raking my brain for hours (so I'm satisfied with some pointers). I know already that the approximation ratio cannot be worse than $2$. I have a wheel graph, where each edge has cost $1$ and the distance between all nodes which are not connected by edges is $2$. The wheel graph $W_6$ is this one:
I have marked in blue what I believe to be the output of an MST heuristic algorithm. But I also think this is the optimal solution, since all nodes can only be visited once. So the cost of the tour would be $7$ for both optimal and MST.
I do not see how this type of graph shows that the $2$-approximation bound of MST heuristic is tight (not necessarily this instance, but the graphs $W_n$ in general). Can someone enlighten me?