It is known that if a plane graph admits a graceful (resp. near-graceful) labeling, then its semidual admits a conservative (resp. near-conservative) labeling. Consequently, we studied gracefulness of two nested cycles graphs considering two different perspectives: the first one by finding graceful (near-graceful) labelings of two nested cycles graphs, and the other one by finding conservative (near-conservative) labelings of its semidual. In this work we prove that for a given integer m 1 ≥ 3 , there exists an integer m ∗ > m 1 such that for all m 2 ≥ m ∗ , if m 1 + m 2 ≡ 0 or 3 ( mod 4 ) (resp. m 1 + m 2 ≡ 1 , 2 ( mod 4 ) ), then there exists a graceful (resp. near-graceful) plane cycle with chords consisting of two nested cycles of sizes m 1 and m 2 , respectively. We also show that the semidual of a plane cycle with chords of size M consisting of two nested cycles is conservative if M ≡ 0 or 3 ( mod 4 ) , and it is near-conservative otherwise.