Definition 6 The dedoubled adjacency graph of G, denoted by, is the graph whose vertices are the adjacencies of G, and for any marker x there is one edge between the vertices (x u) and, and one edge between the vertices (y x) and.

An example of dedoubled adjacency graph is depicted in Fig. 1. In the following, we will simply refer to dedoubled adjacency graphs as adjacency graphs.

Note that all vertices in have degree one or two. Thus, the connected components of are only paths and cycles. These paths and cycles are called elements of .

Given a couple of paralogous markers , an element of the graph is said to contain the couple if it contains the edge linking vertices (x u) and , or the edge linking vertices (y x) and . By definition, a couple can possibly be contained in only one element A of if element A contains both edges and . In this case, A is said to contain twice the couple , and A is called a duplicated element of . If an element A contains no couple twice, then it is called a non-duplicated element of . If the two edges and belong to two different elements A and B of , then A and B will both contain . In this case, we say that A and B intersect. If two elements A and B do not intersect, then we say that A and B are independent. For example in Fig. 1 the two paths of the adjacency graph are duplicated, while the three cycles are non-duplicated. The leftmost path and the leftmost cycle intersect because they both contain the couple , while the two paths are independent. Given an element A of , the set induced by A is the set of couples contained in A.

In this section, we prove the following theorem:

Theorem 1 Let n be the number of couples of paralogous markers in G. Let C_i be the maximum size of a subset of non-duplicated pairwise independent cycles in. The DCJ dedoubling distance of G is d_dcj(G) = n – C_i.

For example, in Fig. 1, the maximum size of a subset of non-duplicated pairwise independent cycles is 2 as there are three cycles, and the two rightmost cycles intersect. The distance would then be d_dcj(G) = 8 – 2 = 6. To prove Theorem 1, we use the following properties:

Property 1 Let n be the number of couples of paralogous markers in G.

1. The maximum size C_i of a set of non-duplicated pairwise independent cycles in the graph is n.

2. If G is dedoubled genome, thencontains n non-duplicated pairwise independent cycles, each containing a single couple of paralogous markers, plus possibly other cycles. In this case, C_i = n.

3. A DCJ operation can only alter the maximum size C_i of a set of non-duplicated pairwise independent cycles by –1, 0 or +1.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

Algorithm 1 is an algorithm that provides a n – C_i length DCJ scenario transforming G into a dedoubled genome.

We now have all the pre-requisites to give the proof of Theorem 1. The proof can be found in Additional file 1 (Supplemental proofs).

Lemma 1 Choosing a maximum size set of non-duplicated pairwise independent cycles inis an APX-complete problem, approximable with an approximation ratio of 2.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

From Lemma 1, the complexity of the Genome Dedoubling problem by DCJ follows immediately.

Corollary 1 The Genome Dedoubling problem by DCJ is NP-complete. Algorithm 1 solves the problem in linear time complexity, except for the computation of the set of cycles S_i that is 2-approximable.

In this section, we search for a minimum length DCJ scenario that transforms a duplicated genome consisting of a single linear chromosome into a dedoubled genome consisting of a single linear chromosome. The results of this section will then be used in the next section for the study of the Genome Dedoubling problem under the reversal model.

In this section and the sequel, G denotes a totally duplicated genome consisting of a single linear chromosome. In this case, the graph contains exactly one path, and possibly several cycles.

Definition 7 The path inis said to be valid if it contains every coupleof paralogous markers in G.

A DCJ operation that merges a cycle c of in the path p is a DCJ operation that acts on an adjacency of c and an adjacency of p, thus gathering c and p into a longer path.

Note that if G is a dedoubled genome, then the path in is necessarily valid. We call such a genome a dedoubled linear genome. So, if the path in is not valid, then any DCJ scenario transforming G into a dedoubled linear genome will merge, in the path, cycles containing the couples that are not contained in the path.

In the following, we always denote by m the minimum number of cycles required to make the path of valid. We also denote by C_i the maximum size of a subset of non-duplicated pairwise independent cycles. First, we have the following property:

Property 2 Let C be the number of cycles in. We have C_i = C – m.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

From Property 2, we then have the following lemma.

Lemma 2 Let n be the number of couples of paralogous markers in G. Let C be the number of cycles in. The minimum length d of a DCJ scenario transforming G into a dedoubled genome consisting of a single linear chromosome equals d = n – C + 2m.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

From Property 2 and Lemma 2, we immediately have the following complexity.

Corollary 2 The problem of finding a DCJ scenario transforming G into a dedoubled genome consisting of a single linear chromosome is NP-complete. Algorithm 1, in which we add the line (2’: Merge in the path all the cycles that are not in S_i) between lines 2 and 3, solves the problem in linear time complexity, except for the computation of the set of cycles S_i that is 2-approximable.

For any couple of paralogous markers in G, the segment of is the smallest segment of G containing both markers x and . For example, in genome , the segment of is , and the segment of is .

Definition 8 The dedoubled overlap graph of G, denoted by, is the graph whose vertices are all the couplesof paralogous markers of G, and there is an edge between two verticesandif the segments of u and v overlap.

An example of dedoubled overlap graph is depicted in Fig. 4.a. In the following, we will simply refer to dedoubled overlap graphs as overlap graphs.

The vertex of the graph is oriented if x and have different signs in G, otherwise it is unoriented. If the vertex is oriented then there exists a reversal operation denoted by that produces the adjacency and a reversal operation denoted by that produces the adjacency . For example, in genome is an oriented vertex of .

The overlap graph of G behaves like arc overlap graphs used in 13 for the Hannenhali-Pevzner theory of sorting by reversal 12. Indeed, given an oriented vertex of the graph , performing the reversal or complements the subgraph induced by and all its neighbouring vertices, and changes the orientation of all vertices in this subgraph (see Fig. 4.b).

A connected component of the graph is oriented if it contains at least one oriented vertex, otherwise it is unoriented. A genome is oriented if all connected components of the graph are oriented, otherwise it is unoriented.

Given an oriented vertex of the graph , the score of is the number of oriented vertices in the overlap graph of the genome obtained after applying on G. Note that the same number of oriented vertices is obtained after applying on G.

Property 3 Let be an oriented vertex of of maximum score. Performing or does not create new unoriented connected components in the overlap graph of the genome obtained.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

In the sequel, we focus on sorting oriented genomes using reversal dedoubling scenarios. A totally duplicated genome G consisting of a single linear chromosome is called a valid-path genome if the single path in is valid. Otherwise, it is called a non-valid-path genome.

In this section, we consider an oriented valid-path genome G. With n being the number of couples of paralogous markers in G, we have the following theorem:

Theorem 2 Let G be an oriented valid-path genome. Let C be the number of cycles in. The reversal dedouhling distance of G is d_rev(G) = n – C.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

In this section, G denotes an oriented non-valid path genome. At least m cycles of have to be merged in the path to make it valid.

An edge or of the adjacency graph is called oriented if markers x and have different signs. Note that extracting a cycle from any element of the graph requires this element to contain oriented edges. It is easy to see that given two adjacencies picked in a given element, a reversal acting on these adjacencies will extract a cycle if and only if the path linking these adjacencies contains an odd number of oriented edges. Thus, we have the following lemma:

Lemma 3 Let G be an oriented non-valid-path genome. Merging a cycle of in its path never creates unoriented connected components in the overlap graph of the genome obtained.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

Theorem 3 Let G be an oriented non-valid-path genome. Let C be the number of cycles in the graphand m be the minimum number of cycles to merge in the path to make it valid. The reversal dedoubling distance of G is d_rev(G) = n – C + 2m.

Proof. See proof in Additional file 1 (Supplemental proofs). ■

Prom Lemma 1 and Property 2, the complexity of the Genome Dedoubling problem by reversal on oriented genomes follows immediately.

Corollary 3 The Genome Dedoubling problem by reversal on oriented genomes is NP-complete. Algorithm 2 solves the problem in quadratic time complexity, except for the computation of S_i that is 2-approximable.