The numbers of 2-connected facially complete planar embeddings on vertices, counted up to homeomorphisms of the sphere,
including reflections and with no distinguished outer
face, are 0, 0, 1, 3, 6, 15, 32, 94, 295, 1169, 4870,
22110, ... for , 2, ... (OEIS A375617;
Tilley et al. 2025).
Chen, Z.-Z.; Grigni, M.; and Papadimitriou, C. "Planarity, Revisited (Extended Abstract)." In Proc. 5th International Workshop on Algorithms
and Data Structures (WADS '97) (Ed. F. Dehne, A. Rau-Chaplin, J.-R. Sack,
and R. Tamassia). Berlin, Germany: Springer-Verlag, pp. 472-473, 1997.
https://doi.org/10.1007/3-540-63307-3.Chen,
Z.-Z.; Grigni, M.; and Papadimitriou, C. H. "Planar Map Graphs." In
Proc. 30th Annual ACM Symposium on Theory of Computing (STOC '98), Dallas, TX,
May 23-26, 1998. (Ed. J. S. Vitter). New York: ACM, pp. 514-523, 1998.
https://doi.org/10.1145/276698.276865.Sloane,
N. J. A. Sequence A375617 in "The
On-Line Encyclopedia of Integer Sequences."Tilley, J.; Wagon, S.;
and Weisstein, E. "A Catalog of Facially Complete Graphs." Util. Math.124,
157-169, 2025. https://doi.org/10.61091/um124-10.
vertices, counted up to homeomorphisms of the sphere,
including reflections and with no distinguished outer
face, are 0, 0, 1, 3, 6, 15, 32, 94, 295, 1169, 4870,
22110, ... for 
, 2, ... (OEIS A375617;
Tilley et al. 2025).
