Before revision:  Our work is the first to exhibit conclusive evidence that Independent Set on C_{\geq k{\sc{Maximum Weight Independent Set}}  -free graphs isnot NP-complete for any integer k. This also generalizes previous work of ours Gartland and Lokshtanov, FOCS '20 , with an additional factor of \log^2(n) in the exponent, where we provided a quasi-polynomial time algorithm for graphs that exclude a path on k vertices as an induced subgraph-free graphs, and by being applicable to a much wider variety of problems, such as }{\sc{Maximum Weight Induced Forest}} {\sc{Maximum Weight Induced Planar Graph}} .
After revision:  Our work is the first to exhibit conclusive evidence that Independent Set on C_{\geq k{\sc{Maximum Weight Independent Set}} -free Gartland and Lokshtanov, FOCS '20 , with an additional factor of \log^2(n) in the exponent, where we provided a quasi-polynomial time algorithm for graphs that exclude a path on k vertices as an induced subgraph-free graphs, and by being applicable to a much wider variety of problems, such as }{\sc{Maximum Weight Induced Forest}} {\sc{Maximum Weight Induced Planar Graph}} .