This paper presents a general technique for obtaining approximation schemes for various NP-complete problems on planar graphs. The strategy involves decomposing a planar graph into k-outerplanar subgraphs, which can be solved optimally using dynamic programming. For maximization problems like maximum independent set, the technique provides a solution that is at least k/(k+1) optimal. For minimization problems like minimum vertex cover, it provides a solution that is at most (k+1)/k optimal. By choosing k as a function of log n, the algorithm achieves polynomial-time approximation with solutions converging toward optimality as n increases. The approach applies to several NP-complete problems, including maximum independent set, minimum vertex cover, and minimum dominating set. The technique also expands the class of planar graphs for which these problems are solvable in polynomial time. The paper includes a detailed algorithm for maximum independent set on outerplanar graphs and extends it to k-outerplanar graphs using dynamic programming. The algorithm efficiently handles the decomposition and merging of subgraphs, ensuring linear-time complexity for k-outerplanar graphs. The results improve upon previous approximation algorithms for these problems, providing better convergence rates and running times.This paper presents a general technique for obtaining approximation schemes for various NP-complete problems on planar graphs. The strategy involves decomposing a planar graph into k-outerplanar subgraphs, which can be solved optimally using dynamic programming. For maximization problems like maximum independent set, the technique provides a solution that is at least k/(k+1) optimal. For minimization problems like minimum vertex cover, it provides a solution that is at most (k+1)/k optimal. By choosing k as a function of log n, the algorithm achieves polynomial-time approximation with solutions converging toward optimality as n increases. The approach applies to several NP-complete problems, including maximum independent set, minimum vertex cover, and minimum dominating set. The technique also expands the class of planar graphs for which these problems are solvable in polynomial time. The paper includes a detailed algorithm for maximum independent set on outerplanar graphs and extends it to k-outerplanar graphs using dynamic programming. The algorithm efficiently handles the decomposition and merging of subgraphs, ensuring linear-time complexity for k-outerplanar graphs. The results improve upon previous approximation algorithms for these problems, providing better convergence rates and running times.