Faster recognition for abstract rotation systems

Determine whether there exists an algorithm with running time strictly better than O(n^5) to decide, given an abstract rotation system of the complete graph K_n, whether the rotation system is generalized twisted.

Background

The paper provides two combinatorial characterizations of generalized twisted drawings and derives algorithms for recognition. Using the five-vertex subrotation characterization, an O(n5)-time algorithm decides generalized twistedness for abstract rotation systems of K_n, while a separate bipartition-based characterization yields an O(n2)-time algorithm for realizable rotation systems.

However, the quadratic-time approach does not directly extend to arbitrary abstract rotation systems, partly due to non-realizability issues (e.g., a non-realizable rotation system of K_5 where the outcome depends on the choice of the vertex used in the procedure). This motivates improving the worst-case complexity for the abstract setting beyond the O(n5) bound.

References

The obvious open question is whether the decision can also be done faster for abstract rotation systems. The quadratic time algorithm we give does not directly apply to them.

— Characterizing and Recognizing Twistedness (2508.16178 - Aichholzer et al., 22 Aug 2025) in Section 6 (Conclusion and open problems)