Fig. 4

For an all-duplicated genome A=(+a+c−b−d)(+a−b)(+c+d) (black edges) and an ordinary genome R=(+a−c−b+d) (blue edges), a) the contracted breakpoint graph \(\hat {\mathfrak {G}}(A,R)\), where the A-cycle is oriented clockwise; b) the polygon \(\mathcal {P}_{o}(A)\) obtained from \(\hat {\mathfrak {G}}(A,R)\), where the blue dashed lines represent a matching on the sides; c) the ht-decomposition of \(\hat {\mathfrak {G}}(A,2R)\) consisting of a single AR-cycle; d) the gluing of \(\mathcal {P}_{o}(A)\) resulting in an embedded graph G _o (A,R) on a 2-torus (with v=1, e=4, f=1)