Let \(\textbf{P}_{2n+2}\) be the regular polygon with \(2n+2\) vertices, and let \(\theta \) be the rotation of 180 \(^\circ \) . Fomin and Zelevinsky proved that \(\theta \) -invariant triangulations of \(\textbf{P}_{2n+2}\) are in bijection with the clusters of cluster algebras of type \(B_n\) and \(C_n\) . Furthermore, cluster variables correspond to the orbits of the action of \(\theta \) on the diagonals of \(\textbf{P}_{2n+2}\) . In this paper, we associate a labeled modified snake graph \(\mathcal {G}_{ab}\) to each \(\theta \) -orbit [a, b], and we get the cluster variables of type \(B_n\) and \(C_n\) which correspond to [a, b] as perfect matching Laurent polynomials of \(\mathcal {G}_{ab}\) . This extends the work of Musiker for cluster algebras of type B and C to every seed.
