Polynomial-time solvability of MWIS for all remaining Alekseev cases

Prove that the Maximum Weight Independent Set problem is solvable in polynomial time on every H-free graph class not covered by the original hardness reductions, namely whenever H is a forest whose connected components each have at most three leaves.

Background

The introduction describes a conjectural boundary for polynomial-time solvability of Maximum Weight Independent Set in H-free graphs. Alekseev's hardness results leave open the cases in which the forbidden graph is a forest and every connected component has at most three leaves.

The cited progress on P_t-free and S_{t,t,t}-free graphs is presented as evidence supporting, rather than proving, this broader conjecture. The statement therefore remains an explicit unresolved conjecture beyond the results established in the paper.

References

This progress corroborates the conjecture that MWIS is polynomial-time solvable in $H$-free graphs for all the open cases left by the original hardness reductions, that is, whenever $H$ is a forest whose every connected component has at most three leaves.

— 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