To extend the authors' approach for cylindrical graphs with cycles of orders not divisible by 3, several modifications and considerations could be made:
Adjusting the Construction of Dominating Sets: The construction of the minimum 2-dominating set could be adapted to account for the specific properties of cycles with orders not divisible by 3. This may involve developing new patterns for selecting vertices that ensure coverage of all vertices in the graph.
Refining the Wasted Domination Technique: The wasted domination technique could be refined to accommodate the unique adjacency relationships present in cycles of different orders. This might involve analyzing the overlap of neighborhoods more closely to minimize wasted domination.
Utilizing Different Matrix Operations: The authors could explore alternative matrix operations or modifications to the tropical matrix product that account for the different cycle structures. This could involve defining new matrices that reflect the adjacency relationships in cycles of varying lengths.
Computational Algorithms: Developing new algorithms that specifically target the characteristics of cycles not divisible by 3 could enhance the computational efficiency and accuracy of determining the 2-domination number in these cases.
Empirical Testing: Conducting empirical tests on various cycle lengths and configurations could provide insights into how the 2-domination number behaves under these conditions, leading to the formulation of new conjectures or theorems.
By implementing these strategies, the authors could effectively extend their analysis to encompass a broader range of cylindrical graphs, enriching the understanding of 2-domination in diverse graph structures.