Before revision: We give an algorithm that takes as input a graph G with weights on the vertices and outputs a maximum weight independent set of G . If G  does not contain any cycle on k or more vertices as an induced subgraph (G is a C _{ \geq k }-free graph ), the algorithm runs in time n^{O( k^3 \log ^5 n) , and therefore for fixed k is a quasi-polynomial time algorithm. C_{\geq 4-free graphs (also known as chordal graphs) have a well known polynomial time algorithm.
After revision: We give an algorithm that takes as input a graph G with weights on the vertices and outputs a maximum weight independent set of G . If G does not contain any cycle on k or more vertices as an induced subgraph (G is a C _{ \geq k }-free graph ), the algorithm runs in time n^{O( k^3 \log ^5 n) -free graphs (also known as chordal graphs) have a well known polynomial time algorithm.