Figure 1

An example of the breakpoint graph and its transformation into an identity breakpoint graph. (a) Graph representation of a two-chromosomal genome P = (+a + b)(+c + e + -d) as two black-obverse cycles and a unichromosomal genome Q = (+a + b - e + c - d) as a red-obverse cycle. (b) The superposition of the genome graphs P and Q. (c) The breakpoint graph G(P, Q) of the genomes P and Q (with removed obverse edges). The black and red edges in G(P, Q) form c(P, Q) = 2 non-trivial black-red cycles and one trivial black-red cycle. The trivial cycle (a^h, b^t) corresponds to a common adjacency between the genes a and b in the genomes P and Q. The vertices in the non-trivial cycles represent breakpoints corresponding to the endpoints of b(P, Q) = 4 synteny blocks: ab, c, d, and e. By Theorem 1, the distance between the genomes P and Q is d(P, Q) = 4 - 2 = 2. (d) A transformation of the breakpoint graph G(P, Q) into the identity breakpoint graph G(Q, Q), corresponding to a transformation of the genome P into the genome Q with two 2-breaks. The first 2-break transforms P into a genome P' = (+a + b)(+c d - e), while the second 2-break transforms P' into Q. Each 2-break increases the number of black-red cycles in the breakpoint graph by one, implying this transformation is shortest (see Theorem 1).