Complexity dichotomies for matching-cut problems on H-free graphs

Complete the complexity dichotomies for Matching Cut and Minimum Matching Cut on H-free graphs, particularly for the remaining graphs H whose connected components are paths or subdivided claws.

Background

The paper establishes polynomial-time algorithms and NP-hardness results for several H-free graph classes, yielding partial complexity classifications for Matching Cut and Minimum Matching Cut. The remaining unresolved cases form a constant-size collection of graphs H, all of whose connected components are paths or subdivided claws. A complete dichotomy would determine, for every graph H, whether each problem is polynomial-time solvable or NP-hard on H-free graphs.

References

In both cases, the computational complexity remains open only for a constant number of graphs $H$. In all open cases, every connected component of $H$ is either a path or a subdivided claw. This leads to the following natural open problem.

Open Problem 1 Complete the complexity dichotomies for and on $H$-free graphs.

— Finding Minimum Matching Cuts in $H$-free Graphs and Graphs of Bounded Radius and Diameter  (2502.18942 - Lucke et al., 26 Feb 2025) in Section Conclusion, Open Problem 1