Fig. 2

Intergenic graphs \(I(\pi ,r^\pi ,r^\iota )\), \(I({\pi '},r^{\pi '},r^\iota )\), and \(I(\iota ,r^\iota ,r^\iota )\), with \(\pi = (3~1~2~4~5~7~6)\), \(r^\pi = (15,\) 6, 4, 12, 8, 13, 9, 2), \(\pi ' = (1~3~2~4~5~7~6)\), \(r^{\pi '} = (10,6,9,12,8,13,9,2)\), \(\iota = (1~2~3~4~5~6~7)\), and \(r^\iota = (10,15,8,7,5,9,13,2)\). Black squares represent intergenic vertices, and the number inside it indicate their sizes. Rounded rectangles in blue represent components. Note that in (a) there are three edges in \(I(\pi ,r^\pi ,r^\iota )\), and \(C(I(\pi ,r^\pi ,r^\iota )) = 2\), and \(C_{odd}(I(\pi ,r^\pi ,r^\iota )) = 2\) since there are five intergenic vertices in \(c_1\) and three intergenic vertices in \(c_2\). We also have in (a) all values for \((\ell (r^\pi _i) - \ell (r^\iota _i))\) and \(\Delta _i(r^\pi ,r^\iota )\), with \(1 \le i \le 8\). The instance \((\pi ',r^{\pi '})\) is the result of applying \(\rho (1,2,8,2)\) to \((\pi ,r^\pi )\). In (b) we can see that compared to \(I(\pi ,r^\pi ,r^\iota )\), \(I(\pi ',r^{\pi '},r^\iota )\) has one more component, and \(e_1\) was removed. In (c) we can see that when we reach the target instance \((\iota ,r^\iota )\) the number of components is equal to the number of intergenic regions in \(\iota\) (i.e., \(C(I(\iota ,r^\iota ,r^\iota )) = n+1 = 8\))