Fig. 3

The unique optimal completion C of BP(A,B) from Fig. 2, where A-open (B-open) vertices are joined by green (blue) double edges, closing the AA-path and linking both AB-paths, which makes d DCJ ind(A,B)=n−c=3. The orange edges M ^′={1^h2^t,2^h4^h, 4^t3^h,3^t1^t,5^t5^h} form a set of non-crossing chords covering all vertices of C. By Claim 2, M ^′ leads to an intermediate genome. Notice that S={5^t5^h} is an artificial singleton, that is, a circular chromosome with only unique genes of A. Therefore, M=M ^′−S={1^h2^t,2^h4^h,4^t3^h,3^t1^t}, representing the circular chromosome (1,2,−4,−3), is an intermediate genome. M is present in the optimal scenario \(\mathcal {S}=\{M_{0}=A, M_{1}=(1,{-3})\), M ₂=(1,2,−4,−3),M ₃=B}, composed by one deletion, one insertion, and one reversal (DCJ)