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.
After revision:  Our work is the first to exhibit conclusive evidence that Independent Set on C_{\geq k{\sc{Maximum Weight Independent Set}} problem on P_t -free graphs isnot NP-complete for any integer k.