I understand that the 3-Opt Heuristic for solving the Traveling Salesman problem involves removing three edges from a graph and adding three more to recomplete the tour. However, I've seen many papers that mention that when three edges are removed, there remain only 2 possible ways to recombine the tour - this doesn't make sense to me.
For example, I found a paper  that says:
The 3-opt algorithm works in a similar fashion, but instead of removing two edges we remove three. This means that we have two ways of reconnecting the three paths into a valid tour1 (ﬁgure 2 and ﬁgure 3). A 3-opt move can actually be seen as two or three 2-opt moves.
However, I count 3 different ways to reconnect the tour. What am I missing here?
Also, can someone link me to an algorithm for 3-opt if possible? I'm just trying to understand it, but I haven't come across any clear algorithms yet: all resources I find simply say "remove three edges, reconnect them". That's it, which is sort of vague.
Here are the 3 tours that seem to me to be 3-opt moves after removing three edges.
- Heuristics for the Traveling Salesman Problem by C. Nilsson