Polynomial-time MWIS algorithm for P_7-free graphs

Determine whether the Maximum Weight Independent Set problem is solvable in polynomial time on P_7-free graphs.

Background

The paper reviews a progression of polynomial-time algorithms for Maximum Weight Independent Set on P_t-free graphs: the problem was solved for P_5-free graphs and subsequently for P_6-free graphs. The next case, P_7-free graphs, is explicitly identified as unresolved.

Although quasipolynomial-time algorithms are known for P_t-free graphs, the paper records polynomial-time solvability for P_7-free graphs as an open case.

References

The case of $P_7$-free graphs remains open.

— Graphs with no long claws: An improved bound for the analog of the Gyárfás' path argument  (2501.13907 - Bourneuf et al., 23 Jan 2025) in Section 1, Introduction