A facially complete planar embedding (Tilley et al. 2024) 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; Tiley et
al. 2024).
Sloane, N. J. A. Sequence A375617 in "The On-Line Encyclopedia of Integer Sequences."Chen, Z.;
Grigni, M.; and Papadimitiou, C. "Planarity, Revisited (Extended Abstract)."
In Proc. 5th WADS, pp. 472-473, 1997.Chen, Z.; Grigni, M.;
and Papadimitiou, C. "Planar Map Graphs." In Proc. 30th STOC., pp. 514-523,
1998.Tilley, J.; Wagon, S.; and Weisstein, E. "A Catalog of Facially
Complete Graphs." 17 Sep 2024. https://arxiv.org/abs/2409.11249.
vertices for 
, 2, ... are given by 0, 0, 1, 3, 6, 15, 32, 94, 295, 1169,
4870, 22110, ... (OEIS A375617; Tiley et
al. 2024).
