A cut performed on a genome A separates two adjacent markers of A. A cut affects a single adjacency v in A: it is done between two symbols of v, creating two open ends. A double-cut and join or DCJ applied on a genome A is the operation that performs cuts in two different adjacencies in A, creating four open ends, and joins these open ends in a different way. In other words, a DCJ rearranges two adjacencies in A, transforming them into two new adjacencies.

Consider a DCJ ρ applied to adjacencies v ₁ = γ ₁ ℓ ₁ ℓ ₄ γ ₄ and v ₂ = γ ₃ ℓ ₃ ℓ ₂ γ ₂, which creates x ₁ = γ ₁ ℓ ₁ ℓ ₂ γ ₂ and x ₂ = γ ₃ ℓ ₃ ℓ ₄ γ ₄. We represent such an operation as ρ = ({γ ₁ ℓ ₁|ℓ ₄ γ ₄, γ ₃ ℓ ₃|ℓ ₂ γ ₂} → {γ ₁ ℓ ₁|ℓ ₂ γ ₂, γ ₃ ℓ ₃|ℓ ₄ γ ₄}). The two adjacencies v ₁ and v ₂ are called the sources, while the two adjacencies x ₁ and x ₂ are called the resultants of ρ 8 . One or more labels among ℓ ₁, ℓ ₂, ℓ ₃ and ℓ ₄ can be equal to ε (the empty string), as well as one or more extremities among γ ₁, γ ₂, γ ₃ and γ ₄ can be equal to ○ (a telomere), A DCJ operation can correspond to several rearrangement events, such as an inversion, a translocation, a fusion or a fission 2 .

Given two genomes A and B, the adjacency graph AG(A, B) 3 is the bipartite multigraph whose vertices are the adjacencies of A and of B and that has one edge for each common extremity of a pair of vertices. The graph AG(A, B) is composed of connected components that alternate vertices in genome A and in genome B. Each component can be either a cycle, or an AB-path (which has one endpoint in genome A and the other in B), or an AA-path (which has both endpoints in genome A), or a BB-path (which has both endpoints in B). A special case of an AA or a BB-path is a linear singleton, that is a linear chromosome represented by an adjacency of type ○ℓ○. In Figure 3 we show the example of an adjacency graph.

Components with 3 or more vertices need to be reduced - by applying DCJ operations - to components with only 2 vertices, that can be cycles or AB-paths 8 . This procedure is called DCJ-sorting of A into B. The number of AB-paths in AG(A, B) is always even and a DCJ operation can be of three types 5 : it can either increase the number of cycles by one, or the number of AB-paths by two (optimal DCJ); or it does not affect the number of cycles and AB-paths (neutral DCJ); or it can either decrease the number of cycles by one, or the number of AB-paths by two (counter-optimal DCJ). We assign the same cost to any DCJ operation. For simplicity, we consider the DCJ cost equal to one. Then, the DCJ distance of A and B, denoted by d_DCJ (A, B), corresponds to the minimum number of steps required to do a DCJ-sorting of A into B and is given by the following theorem.

Theorem 1([3]) Given two genomes A and B without duplicated markers, we have d D C J ( A , B ) = | G | - c - b 2 , where   G is the set of common markers and c and b are, respectively, the number of cycles and of AB-paths in AG (A, B).

In order to deal with unique markers, we need operations that change the content of a genome. These operations can be an insertion or a deletion of a block of contiguous markers. Insertions and deletions can be jointly called indel operations. We consider a model in which an indel only affects the label of one single adjacency, by deleting or inserting contiguous markers in this label, with the restriction that an insertion cannot produce duplicated markers 5 . In other words, while sorting A into B, the indel operations are the steps in which the markers in   A are deleted and the markers in   B are inserted.

Given ℓ ₃ ≠ ε, the deletion of ℓ ₃ from the adjacency γ ₁ ℓ ₁ ℓ ₃ ℓ ₂ γ ₂ is represented as (γ ₁ ℓ ₁|ℓ ₃|ℓ ₂ γ ₂ → γ ₁ ℓ ₁|ℓ ₂ γ ₂), while the insertion of ℓ ₃ in the adjacency γ ₁ ℓ ₁ ℓ ₂ γ ₂ is represented as (γ ₁ ℓ ₁|ℓ ₂ γ ₂ → γ ₁ ℓ ₁|ℓ ₃|ℓ ₂ γ ₂). One or both extremities among γ ₁ and γ ₂ can be equal to ○ (a telomere), as well as one or both labels among ℓ ₁ and ℓ ₂, can be equal to ε (the empty string). Observe that at most one chromosome can be entirely deleted or inserted at once. Moreover, since duplications are not allowed, an insertion of a marker that already exists is not allowed. Consequently, in this model, it is not possible to apply insertions and/or deletions involving the markers in   G .

Given two genomes A and B, the DCJ-indel distance of A and B, denoted by d D C J i d ( A , B ) , is the minimum cost of a DCJ-indel sequence of operations which sorts A into B, assigning the cost of 1 to each DCJ and a positive cost w ≤ 1 to each indel operation. If w = 1, the DCJ-indel distance corresponds exactly to the minimum number of steps required to sort A into B 5 .

Let us recall the concept of run, introduced by Braga et al. 5 . Given two genomes A and B and a component C of AG(A, B), a run is a maximal subpath of C, in which the first and the last vertices are labeled and all labeled vertices belong to the same genome (or partition). A run is then a subpath of a component and can be represented by its list of vertices. A vertex v that corresponds to an entire run is called a compact-run. If a run is not compact, it is a long-run. An example of a component with 3 runs is given in Figure 4. A run in genome A is also called an   A -run, and a run in genome B is called a   B -run. We denote by Λ(C) the number of runs in a component C. While a path can have 0 or any positive number or runs, a cycle has either 0, 1, or an even number of runs.

A set of labels of one genome can be accumulated with DCJs. In particular, when we apply optimal DCJs on only one component of the adjacency graph, we can accumulate an entire run into a single adjacency 5 . It is possible to do a separate DCJ-sorting using only optimal DCJs in any component C of AG(A, B) 8 . We denote by d_DCJ (C) the number of optimal DCJ operations used for DCJ-sorting C separately (d_DCJ (C) depends only on the number of vertices or, equivalently, the number of edges of C 8 ). The DCJ distance can also be re-written as d_DCJ (A, B) = ∑ _{c∈AG(A, B)} d_DCJ (C).

Runs can be merged by DCJ operations. Consequently, during the optimal DCJ-sorting of a component C, we can reduce its number of runs. The indel-potential of C, denoted by λ(C), is defined by Braga et al. 5 as the minimum number of runs that we can obtain doing a separate DCJ-sorting in C with optimal DCJ operations. An example is given in Figure 5.

The indel-potential of a component depends only on its number of runs:

Proposition 1 ( 5 ) Given two genomes A and B and a component C of AG(A, B), the indel-potential of C is given by λ ( C ) = Λ ( C ) + 1 2 , if Λ(C) ≥ 1. Otherwise, if Λ(C) = 0, then λ(C) = 0.

Let λ ₀ and λ ₁ be, respectively, the sum of the indel-potentials for the components of the adjacency graph before and after a DCJ operation ρ, and let Δλ(ρ) = λ ₁ - λ ₀. If ρ is an optimal DCJ acting on two adjacencies of a single component of the graph, the definition of indel-potential implies Δλ(ρ) ≥ 0. We also know that Δλ(ρ) ≥ 0, if ρ is counter-optimal, and Δλ(ρ) ≥ -1, if ρ is neutral 5 . This allows us to exactly compute d D C J i d ( C ) , that is the DCJ-indel distance of a component C of A G ( A , B ) : d D C J i d ( C ) = d D C J ( C ) + w λ ( C ) 6 . We can then derive the following upper bound for the DCJ-indel distance:

Lemma 1 ( 6 ) Given two genomes A and B without duplicated markers and a positive indel cost w ≤ 1, we have

d D C J i d ( A , B ) ≤ d D C J ( A , B ) + w ∑ C ∈ A G ( A , B ) λ ( C ) .

A DCJ that accumulates labels is always applied to two labeled adjacencies and results into a clean adjacency and an adjacency containing the concatenation of the labels of the original adjacencies. In general, we can represent such an accumulating DCJ ρ by ({γ ₁ ℓ ₁|γ ₄, γ ₃|ℓ ₂ γ ₂}→{γ ₁ ℓ ₁|ℓ ₂ γ ₂, γ ₃|γ ₄}). If ρ accumulate labels of an   A -run, it is denoted by A A A ≻ B . Similarly, if ρ accumulates labels of a   B -run, it is denoted by B B B ≻ A .

The inverse of an accumulating DCJ ρ is a splitting DCJ ρ ^-1 = ({γ ₁ ℓ ₁|ℓ ₂ γ ₂, γ ₃|γ ₄} → {γ ₁ ℓ ₁|γ ₄, γ ₃|ℓ ₂ γ ₂}). Observe that, if ρ is applied on A, ρ ^-1 is applied on B and split a label of an   A -run. In other words, the inverse of an A A A ≻ B is a DCJ applied on B that separates vertices belonging to the same   A -run in two different cycles, denoted by B ≺ A A A . Similarly, the inverse of a B B B ≻ A is a DCJ applied on A that separates vertices belonging to the same   B -run, denoted by A ≺ B B B . An A ≺ B B B or a B ≺ A A A is called an inverted-split. In Table 1 we summarize the operations described above.

Let n be a positive integer, such that n ≥ 2 and let r ₁ = v ₁ x ₁ v ₂ x ₂ ... v_ix_i ... v_jx_j ... v _n-1 x _n-1 v_n be a long-run, in which v ₁ and v _n are labeled, each v_k (2 ≤ k ≤ n - 1) can also be labeled and all x_k (1 ≤ k ≤ n - 1) are clean. We say that two vertices v _i and v _j (1 ≤ i <j ≤ n) in r ₁ are partners if v _i and v _j are labeled and all vertices between v _i and v _j in r ₁ are clean. We can apply an accumulating DCJ on the two partners v _i and v_j , accumulating their labels into a new vertex v _i-j , reducing r ₁ to r ₂ = v ₁ x ₁ v ₂ x ₂ ... v _i-1 x _i-1 v _i-j x _j v _j+1 x _j+1 ... v _n-1 x _n-1 v_n . The subsequent step of accumulation then occurs between two partners of r ₂, reducing r ₂ to r ₃, and so on. Assuming that the initial r ₁ has m ≤ n labeled vertices, we need to apply m - 1 accumulating operations. In the end of the process, we obtain the compact-run r_m , that corresponds to a single vertex whose label is the accumulation of all labels of r ₁. Observe that all labeled vertices will be used in some accumulating DCJ, until the compact-run r_m is obtained.

As an example, take v ₁ = γ ₁ ℓ ₁ γ ₂, x ₁ = γ ₂ γ ₃, v ₂ = γ ₃ ℓ ₂ γ ₄, x ₂ = γ ₄ γ ₅, v ₃ = γ ₅ ℓ ₃ γ ₆, x ₃ = γ ₆ γ ₇, v ₄ = γ ₇ ℓ ₄ γ ₈, with all ℓ _k ≠ ε and let r ₁ = v ₁ x ₁ v ₂ x ₂ v ₃ x ₃ v ₄ be a   B -run. We can start the accumulation with a DCJ of type B B B ≻ A on partners v ₂ and v ₃, creating v _2-3 = γ ₃ ℓ ₂ ℓ ₃ γ ₆ and γ ₄ γ ₅, reducing r ₁ to r ₂ = v ₁ x ₁ v _2-3 x ₃ v ₄. We then apply another DCJ of type B B B ≻ A on partners v ₁ and v _2-3, creating v _1-2-3 = γ ₁ ℓ ₁ ℓ ₂ ℓ ₃ γ ₆ and γ ₂ γ ₃, reducing r ₂ to r ₃ = v _1-2-3 x ₃ v ₄. Finally, we apply a DCJ of type B B B ≻ A on partners v _1-2-3 and v ₄, creating v _1-2-3-4 = γ ₁ ℓ ₁ ℓ ₂ ℓ ₃ ℓ ₄ γ ₈ and γ ₆ γ ₇, reducing r ₃ to r ₄ = v _1-2-3-4. If we follow the accumulation of a run, considering only the labeled vertices, we obtain a rooted tree that is built from the leafs to the root (see Figure 8). The root of the tree indicates the possible positions of a deletion.

The inversion of the run accumulation described in the example above is the inverted-split of the label of the compact-run r ₄ = v _1-2-3-4 into the labeled vertices v ₁ = γ ₁ ℓ ₁ γ ₂, v ₂ =γ ₃ ℓ ₂ γ ₄, v ₃ = γ ₅ ℓ ₃ γ ₆ and v ₄ = γ ₇ ℓ ₄ γ ₈. We start by applying a B ≺ A A A DCJ on v _1-2-3-4 = γ ₁ ℓ ₁ ℓ ₂ ℓ ₃ ℓ ₄ γ ₈ and γ ₆ γ ₇, obtaining v _1-2-3 = γ ₁ ℓ ₁ ℓ ₂ ℓ ₃ γ ₆ and v ₄ = γ ₇ ℓ ₄ γ ₈. We then apply a B ≺ A A A on v _1-2-3 and γ ₂ γ ₃, obtaining v ₂ = γ ₁ ℓ ₁ γ ₁ and v _2-3 = γ ₃ ℓ ₂ ℓ ₃ γ ₆. Finally we apply a B ≺ A A A on v _2-3 and γ ₄ γ ₅, obtaining v ₂ = γ ₃ ℓ ₂ γ ₄ and v ₃ =γ ₅ ℓ ₃ γ ₆. If we follow the inverted-split of a run, considering only the labeled vertices, we obtain a rooted tree that is built from the root to the leafs (see Figure 8 again). In this case, the first inverted-split defines the root. Then, each one of the subsequent inverted-splits must be chained with a DCJ in this tree. The root of the tree indicates the possible positions of an insertion.

Disregarding recombinations, we can first perform the genome capping, a technique that helps us to avoid difficulties and special cases produced by telomeres: we adjoin new markers (caps) to the ends of the chromosomes (and new chromosomes composed of caps only, if necessary) so that we do not change the distance and we do not have to worry about telomeres 4 . After the capping, the two genomes have the same number of chromosomes and the corresponding adjacency graph contains only clean paths of size 1 and cycles. Recall that, since AG(A, B) is bipartite, all cycles have even length and can have 0, 1 or an even number of runs. Capped genomes can be then sorted with translocations (which mimic also fusions and fissions), inversions, circular chromosome excisions and reincorporations.

An important step of the DCJ-indel sorting is to merge runs in cycles with at least 4 runs, so that the indel-potential for each cycle is achieved.

Proposition 5 The indel-potential of a cycle C with at least 4 vertices and 2 or more runs can be achieved by extracting from C a cycle with a single run.

Proof: For any positive integer i let λ ( i ) = i + 1 2 . If Λ(C) = 2, we can split C into two cycles containing a single run each, and the indel-potential is preserved. For any cycle C with 4 or more runs, since the number of runs in this case is always even, we have λ ( i ) = i 2 + 1 . We then denote by λ' the alternative potential, obtained by extracting cycles with a single run from C. Observe that, for any i = 4, 6, 8, ..., λ'(i) = λ(i - 2) + 1. It is easy to check the base case, that is λ ′ ( 4 ) = λ ( 2 ) + 1 = 2 + 1 = 3 = 4 2 + 1 = λ ( 4 ) . By induction, for i = 6, 8, 10 ..., we have λ ′ ( i ) = λ ( i - 2 ) + 1 = i - 2 2 + 1 + 1 = i 2 + 1 = λ ( i )    □

In the restricted sorting of linear genomes a circular chromosome has to be immediately reincorporated after its excision - these two consecutive operations mimic either a transposition or a block-interchange 2 4 . As we have seen before, the general DCJ-indel sorting is bi-directional - the operations can be applied on genome A or B, depending on whether we accumulate runs in A or in B. However, when a DCJ creates a circular chromosome, we need to apply the subsequent DCJ on the same genome, and it is not easy to see how this interferes with the indel-potential of AG(A, B).

Suppose that a DCJ performed an excision of a circular chromosome. Let (v ₁, v ₂) be a pair of vertices, such that v ₁ and v ₂ are in the same genome and belong to the same cycle in AG(A, B), v ₁ is an adjacency at the circular chromosome and v ₂ is an adjacency at a linear chromosome. The pair (v ₁, v ₂) is called a link. Since v ₁ and v ₂ are in the same cycle, a chromosome reincorporation can always be done by applying a DCJ on the two vertices v ₁ and v ₂ 8 .

The cycle to which a link (v ₁, v ₂) belongs is called a connection cycle. Let C be a connection cycle of AG(A, B) with 2k ≥ 4 vertices. Since C has k vertices in each genome, there are at least k - 1 and at most k 2 . k 2 distinct links in C.

The two vertices v ₁ and v ₂ of a link in a connection cycle C are connected by two distinct subpaths of C. The distance between v ₁ and v ₂ is given by the number of edges in the shortest path connecting them. Since both v ₁ and v ₂ are in the same genome, this distance is always even and positive. If the distance between v ₁ and v ₂ is 2, v ₁ and v ₂ have a common neighbor, and (v ₁, v ₂) is called a short-link.

Proposition 6 After the excision of a circular chromosome by a DCJ, there is at least one short-link in AG(A, B).

Proof: Suppose that the circular chromosome is in genome A. If AG(A, B) contained no connection cycle, genome B would also have a circular chromosome, which would be a contradiction. Let C = v ₁ x ₁ v ₂ x ₂ ... v _n x _n be a connection cycle in AG(A, B), in which the vertices v ₁, ..., v _n are in A and the vertices x ₁, ..., x _n are in B, and let (v_i, v_j ) be a link in C such that v _i is in the circular chromosome and v _j is in a linear chromosome of A. Consider without loss of generality that i <j. Then take the vertex v_k, i ≤ k <j, such that k is the largest index of a vertex between v _i and v _j belonging to the circular chromosome. Then (v_k, v _k+1) is a short-link.    □

In order to find out whether the indel-potential of the connection cycle C can be preserved after applying a DCJ on a certain link (v ₁, v ₂), basically we need to analyze how the connection cycle C is split, by analyzing the vertices that are between v ₁ and v ₂ in C.

We focus on the short-links only. Let (v ₁,v ₂) be a short-link in a connection cycle C, such that v ₁ = γ ₁ ℓ ₁ γ ₂ and v ₂ = γ ₃ ℓ ₂ γ ₄ (ℓ ₁ and ℓ ₂ can be equal to ε). Without loss of generality, let z = γ ₂ ℓ ₃ γ ₃ be the common neighbor of v ₁ and v ₂ (ℓ ₃ can also be equal to ε). We then define the optimal DCJ ρ(v ₁, v ₂) = ({v ₁, v ₂} → {x ₁, x ₂}), such that x ₁ = γ ₂ γ ₃ and x ₂ = γ ₁ ℓ ₁ ℓ ₂ γ ₄. Observe that ρ(v ₁, v ₂) always extracts z together with a new clean vertex x ₁ into a cycle, and accumulates the labels of v ₁ and v ₂ into a new vertex x ₂, which is extracted into a cycle with the remaining vertices of C. There are three different cases:

1. Gaps: If the two vertices of a short-link have a clean common neighbor, it is called a gap. A DCJ applied to a gap of a connection cycle C splits C into a clean cycle C' and a cycle C'' that has the same indel-potential of C.

2. Compact-runs: Let (v ₁, v ₂) be a short-link in AG(A, B), such that the common neighbor z of v ₁ and v ₂ is a compact-run. An optimal DCJ ρ(v ₁, v ₂) extracts the compact-run z and a new clean vertex into a new cycle. According to Proposition 5, ρ(v ₁, v ₂) preserves the indel-potential of AG(A, B).

3. Inverted-splits: If a short-link (v ₁, v ₂) is not a gap nor is separated by a compact-run, only one possiblity remains: the common neighbor z of v ₁ and v ₂ is labeled and belongs to a long-run r. Observe that an optimal DCJ ρ(v ₁, v ₂) splits C into a cycle C' containing a new clean vertex and z (Λ(C') = 1) and a cycle C'' containing all remaining runs of C and the remaining vertices of r, that is, we have Λ(C'') = Λ(C).

Although the overall indel-potential seems to be increased, the DCJ described above is an inverted-split of type A ≺ B B B if the circular chromosome is in A and r is in B (or, symmetrically, of type B ≺ A A A , if the circular chromosome is in B and r is in A). We have seen that inverted-splits, if properly applied, do the backtracing of the insertion position of a run in the opposite genome and do not increase the indel-potential of AG(A, B).

It is important to guarantee that, after applying a DCJ that inversely splits a run r ₁ and another DCJ that inversely splits another run r ₂, the runs r ₁ and r ₂ are not merged. We do this by simply extracting the residual part of an inversely split run into a new cycle. Furthermore, during the merging or accumulation of runs, a run r can be inversely split by successive DCJs. In this case, we need to guarantee that each new inverted-split of r is either the first or chained with one of the previous inverted-splits.

We can always reincorporate the circular chromosome with a DCJ applied to any short-link (v ₁, v ₂), except if ρ(v ₁, v ₂) splits a run r that is already inversely split and ρ(v ₁, v ₂) cannot be chained with a previous inverted-split of r. However, in this case, r will be separated alone in a cycle (each run is immediately separated after its first inverted-split).

After an excision, suppose that the circular chromosome is in genome A (respectively in B). Let C be a connection cycle in AG(A, B). For each vertex v of C in A (respectively in B), there is at least one link containing v. Due to this fact, when we have a cycle containing a single inversely split run r, it is easy to find a link chained with a previous inverted split of r.

Proposition 7 If a connection cycle C with a single run r has links in one genome and its run r is in the other genome, we can always reincorporate the circular chromosome and preserve the indel-potential.

Proof: Let C have links in genome A. Each short-link of C is either a gap, or a compact-run, or the first inverted-split of the   B -run r. Otherwise, C has in genome A a vertex v that was created by a previous inverted-split ρ of r. Since each vertex of C in A is part of a link, we can choose a link that contains v and, consequently, is chained with ρ.    □

We put everything together in Algorithm 1 (Additional file 1) and describe the sorting of capped genomes for the restricted model, in which each circular chromosome is reincorporated immediately after its creation. Applying this procedure we can find a sequence of optimal DCJs that sort A into B while preserving the indel-potential. In other words, this algorithm results in a sorting sequence in the restricted model that has exactly the same cost given by the upper bound of Lemma 1.