Yes, the Hamiltonian cycle condition based on Sk-MaxPSS-disjointness can indeed be generalized to other types of spanning subgraphs or connectivity requirements beyond strong connectivity. The core idea behind the condition is the relationship between the existence of certain spanning subgraphs and the disjointness of maximal proper subsets.
For instance, one can explore the conditions under which a graph is Hamiltonian based on different levels of edge connectivity, such as k-edge-connectedness. By establishing a similar framework to that of Sk-MaxPSS-disjointness, we can define a new set of conditions that relate to the existence of Hamiltonian cycles in k-edge-connected graphs. This would involve analyzing the structure of the graph and the properties of its edge cuts, ensuring that the subsets generated maintain the required connectivity.
Furthermore, this generalization can extend to other types of connectivity, such as vertex connectivity or even more complex structures like biconnected components. By applying the principles of SSD systems, one can derive conditions for the existence of cycles or paths that visit all vertices while adhering to the specified connectivity requirements.
In summary, the Hamiltonian cycle condition can be effectively generalized to encompass a broader range of connectivity requirements, leveraging the foundational principles established in the study of SSD systems and their applications to graph theory.