A characteristic string approach is presented for derivation and isomorphism identification of valid topological graphs (TGs) for some 1, 2-DOF (degree of freedom) planar closed mechanisms from the contracted graphs (CGs). First, topological graph (TG) is simplified by the topology graph with digits (DTG) for the 1-, 2-DOF planar closed mechanisms. Second, the equivalent conditions between the characteristic string and DTG are determined and verified by some created simulation mechanisms of valid TGs, a complicated derivation of TG is transformed into an easy derivation of the characteristic string. Third, the valid characteristic strings are derived corresponding to non-isomorphic TGs, and the invalid characteristic strings corresponding to the isomorphic and invalid TGs are determined using the characteristic string. Finally, many valid characteristic strings for the 2-DOF planar mechanisms are derived from some valid characteristic strings for the 1-DOF planar mechanisms.