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. A subexponetial time algorithm for C_{\geq 5}-free graphs (also known as long-hole-free graphs) was found in 2019 }  [ Chudnovsky et al., Arxiv'19 ] followed by a polynomial time algorithm for C_{\geq 5 -free graphs in 2020 [ Abrishami et al. , Arxiv'20 ] . For k > 5 only a quasi-polynomial time approximation scheme Chudnovsky et al., SODA'20%DIFDELCMD < ] %%% was known. 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%DIFDELCMD < ]%%% , 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}} .
For an integer t, a graph G is called if G does not contain any induced cycle on more than~t vertices. We prove the following statement: for every pair of integers d and t and a CMSO_2 statement~\phi, there exists an algorithm that, given an n-vertex C _{ >t }-free graph G with weights on vertices, finds in time n^{O( \log ^4 n) -free graphs (also known as chordal graphs) have a well known polynomial time algorithm. A subexponetial time algorithm for C_{\geq 5}-free graphs (also known as long-hole-free graphs) was found in 2019 } a maximum-weight vertex subset S such that G [ S ] has degeneracy at most d and satisfies \phi. The running time can be improved to n^{O(\log^2 n) -free , that is, G does not contain an induced path on t vertices. This expands the recent results of the authors [ to appear at FOCS 2020 and SOSA 2021 ] %DIFDELCMD < ] %%% on the{\sc{Maximum Weight Independent Set}} problem on P_t -free %DIFDELCMD < ]%%% graphs in two directions: by encompassing the more general setting of C_{>t-free graphs, and by being applicable to a much wider variety of problems, such as }{\sc{Maximum Weight Induced Forest}} or{\sc{Maximum Weight Induced Planar Graph}} .
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