[Q8] Genome rearrangements: double-cut-and-join distance

Let b and c be the number of breakpoints and cycles, respectively, as pointed by the breakpoint graph G for two genomes. Consider the next cases for the DCJ operation:

1. Cutting two black lines belonging to different cycles and then joining
2. Improper joining
3. Proper joining with no successive lines
4. Proper joining with successive lines in a cycle with two or more black lines
5. Proper joining with successive lines in a 2-cycle

In which of the cases above, performing the DCJ operation would decrease c?

1. Just in I and V
2. Just in II, IV
3. Just in I, III and V
4. Just in II, III and IV
5. None of the above.

Original idea by: Juan Felipe Hernández Albarracín.