### Disjoining 3-exchanges as base for 5-opt non-sequential moves, part 1

So far we have considered 5-opt non-sequential moves obtained by extending double bridge or crossed bridge move. Double bridge and crossed bridge themselves are obtained from disjoining 2-exchange submove, that is from 2-exchange starting with `12-34`

. But they are other disjoining k-exchanges that we can start from to built an 5-opt non-sequential move.

Among eight possible types of 3-exchanges there are four *disjoining* 3-exchange moves. We have already considered them, though in a different context, namely while examining 3-exchanges during constructing a sequential move. These disjoining 3-exchanges are:

```
... TO_123456 # disconnecting...
... TO_123465 # disconnecting...
... TO_124356 # disconnecting...
... TO_126543 # disconnecting...
```

12-34-56 | ||

12-34-65 | 12-43-56 | 12-65-43 |

The first of these disjoining 3-exchanges, `12-34-56`

, breaks tour into *three* separated cycles. It is not possible to join them into feasible tour by any additional single 2-exchange (to have then 5 exchanges in total). So we would skip this case and take a closer look at the remaining three 3-exchanges, each breaking tour into two cycles only.

General idea of construction of 5-opt non-sequential moves from disjoining 3-exchange is similar to the idea of construction of 4-opt non-sequential moves from disjoining 2-exchange and is shown on diagrams for four descentants of `12-34-65`

below:

12-34-65 | AEcDB | AdCeB |

12-34-65 | ADeCB | AcEdB |

Note that while from one disjoining 2-exchange (`12-34`

) we have obtained two 4-opt non-sequential move types: double bridge (*ADBC*) and crossed vertical bridge (*AcdB*), but any of the three disjoining 3-exchanges gives us *four* 5-opt non-sequential move types. For example `12-34-65`

gives: *AEcDB*, *AdCeB*, *ADeCB*, *AcEdB*.