Graphoidal Covering Numbers of Strong and Enhanced Power Graphs of Cyclic and Dihedral Groups

This study investigates graphoidal coverings of power graphs, enhanced power graphs, and strong power graphs associated with selected finite cyclic and dihedral groups. The analysis focuses on determining graphoidal covering numbers and on clarifying how the underlying group structure and the relevant adjacency relation affect these parameters in a systematic manner. For cyclic groups, the graphoidal covering number of the strong power graph is obtained for the cases n = 2, n = 3, n = 4, prime n ≥ 5, and composite n ≥ 6. The study also characterises the cyclic groups for which the power graph and the strong power graph are isomorphic, showing that, for n > 2, this occurs precisely when n = 2p, with p an odd prime. In addition, the graphoidal covering number of the enhanced power graph of Zn is determined for n ≥ 4. For dihedral groups, the structures of the power, enhanced power, and strong power graphs are examined, and explicit graphoidal coveringnumbers are derived for D2p, where p is an odd prime. The resulting comparison establishes a strict ordering among the three covering numbers in this case. Overall, the results show that variations in group structure and graph adjacency produce distinct graphoidal covering behaviour across the graph classes considered.