Minimum Matching Cut on subcubic graphs

Determine the computational complexity of Minimum Matching Cut on graphs of maximum degree 3.

Background

The paper places its results in the context of complexity questions for matching-cut variants on graphs of bounded maximum degree. Although a related matching-cut variant is known to be NP-hard for graphs of maximum degree 3, every graph with at least seven vertices and maximum degree 3 has a matching cut. The unresolved issue is therefore the complexity of optimizing the size of a matching cut in this subcubic setting.

References

Also, the computational complexity of graphs of bounded maximum degree has been investigated for different variants of . While is known to be NP-hard for graphs of maximum degree~$3$, every graph with at least~$7$ vertices and maximum degree~$3$ has a matching cut. This leads to our second open problem.

Open Problem 2 Determine the complexity of on graphs of maximum degree~$3$.

— 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 2