A graph-theoretical analysis is performed on the (H2O)(20) "edge-sharing pentagonal prism" cluster to find proton configurations that yield the lowest cluster total energies. Using the low-energy structures, we create models for protonated (H2O)(20)H+ clusters that compete energetically with "cage-like" structures proposed earlier in the literature. We perform benchmarking between different theoretical methods and observe significant stabilization of compact versus polyhedral clusters due to long-range electron correlation effects, which make the comparison between different cluster morphologies difficult using density functional theory only. All methodologies that we used (up to the second level of Moller-Plesset perturbation theory (MP2)) agree that for protonated clusters, the cage-like morphology proposed in [Chem. Phys. Lett. 2000, 324, 279-2883 is the most stable one. We study in detail several (H2O)(20)H+ cluster structures and suggest that the energetics in small protonated water clusters is dominated by the competition of open polyhedral structures, favored by the Eigen H9O4+ species, against van der Waals interaction and "ice rules", which both favor compact structural motifs such as the prismatic particle. We demonstrate this tendency using ab initio calculations for prismatic, dodecahedral, and cage-like clusters, while the "magic number" cluster (H2O)(n)H+, n = 21, is observed to minimize all competing energy contributions simultaneously. We emphasize the utility of the cage-like cluster as a model template for ice-related atmospheric reactions and benchmark the GPAW density functional theory code for making such calculations.