Complexity of Threading with turn costs on maximum-degree-3 graphs

Determine whether the Threading problem under the turn-cost metric admits a polynomial-time algorithm on graphs whose maximum vertex degree is at most 3.

Background

The paper introduces Threading under a turn-cost metric and proves several hardness results, including NP-completeness for general graphs (even with 0/1 turn costs) and for graphs of maximum degree 4. In contrast, it gives polynomial-time algorithms for the special cases Perfect Threading and Double Threading on graphs of maximum degree 3.

Despite these advances, the computational complexity of the general Threading problem on graphs of maximum degree 3 remains unresolved. Establishing whether a polynomial-time algorithm exists (or alternatively, a hardness result) would complete the degree-based complexity landscape initiated by the paper's results for degrees 3 and 4.

References

It remains an open question whether the general Threading problem can be solved efficiently in graphs with maximum degree 3.

— Graph Threading with Turn Costs  (2405.17953 - Demaine et al., 2024) in Section 4 (Maximum-Degree-3 Graphs), end