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Facially Complete Planar Embedding


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).

FaciallyComplete2ConnectedGraphs

The numbers of 2-connected facially complete planar embeddings for on n vertices for n=1, 2, ... are given by 0, 0, 1, 3, 6, 15, 32, 94, 295, 1169, 4870, 22110, ... (OEIS A375617; Tiley et al. 2024).


See also

Complete Graph, Cyclic Chromatic Number, Graph Face, Map Graph, Planar Embedding, Planar Graph

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References

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.

Cite this as:

Weisstein, Eric W. "Facially Complete Planar Embedding." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FaciallyCompletePlanarEmbedding.html

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