Figure 3.

Examples when the assumption holds and fails for Theorem 4.2. In this case, 𝒟^*(x_i) = 3 and the assumption in Theorem 4.2 holds for curve R1 but fails for curve R2. The assumption of the modified duplication strategy that $r_i^{(k)}=r_i$ if a_i ∈ {k, k + 1} holds for both curves.