A facially complete planar embedding (Tilley et al. 2025) is a planar embedding that becomes complete when edges
are added for every two vertices that lie on a common face
(and do not already correspond to edges in the original embedding).
The numbers of 2-connected facially complete planar embeddings for on vertices for , 2, ... are given by 0, 0, 1, 3, 6, 15, 32, 94, 295, 1169,
4870, 22110, ... (OEIS A375617; Tilley et
al. 2025).
Chen, Z.; Grigni, M.; and Papadimitiou, C. "Planarity, Revisited (Extended Abstract)." In Proc. 5th WADS, pp. 472-473,
1997.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 for 
, 2, ... are given by 0, 0, 1, 3, 6, 15, 32, 94, 295, 1169,
4870, 22110, ... (OEIS A375617; Tilley et
al. 2025).
