Sequential moves and tour order

Let us take a look at diagrams of two 3-opt sequential moves:

Move A	Move B

In both cases, A and B, the same set of cities is used to make a move: {c1, c2, c3, c4, c5, c6}, or: the same set of positions in tour. However, schemes for each of these two moves are different, and therefore the results would be different.

How can we distinguish between move of type A and of type B then?

We can use simple and unambiguous notation: by indicating the tour order of cities involved, starting from c1.

Let us take move A: we start from c1 and move along the tour clockwise. The next city from the set is c2 (second city engaged in move), then we encounter c4 (that is: fourth city of move), c3 (third city of move), c6 (sixth city of move) and c5 (fifth city of move). The tour order of cities in this sequential move is: c1, c2, c4, c3, c6, c5, so the move can be described as 12-43-65.

Move A = 12-43-65	Move B = 12-56-43

In move B the tour order of cities is: c1, c2, c5, c6, c4, c3, so this move can be described as 12-56-43.

This notation is simple, unambiguous and natural. Thanks to it we no longer need to use some arbitrary numbers as labels for each type of 3-opt move, not saying anything about scheme of given move. Moreover, this notation can be used to identify a scheme of any other sequential move, for example sequential 4-opt moves, 5-opts and so on.

Three examples of 4-opt sequential moves

Move 12-87-43-65	Move 12-78-56-43	Move 12-43-65-87