2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160233We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2-pole networks for the edges of core graphs. Using structure theorems for toroidal and projective-planar graphs containing no K33-subdivisions, we apply these techniques to obtain their unlabelled enumeration.25 pages (some corrections), 4 figures (one figure added), 3 tablesCombinatoricsDiscrete Mathematics05A15, 05C30 (Primary); 05C70, 05C75, 05C38 (Secondary)text