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# Critical Nonplanar Graph

A nonplanar graph is said to be critical nonplanar if the removal of a vertex results in a planar graph for every vertex of .

Critical nonplanar graphs differ from apex graphs in that an apex graph requires only that there exist at least one vertex whose removal gives a planar graph, while a critical nonplanar graph requires that removal of each vertex gives a planar graph.

Classes of graphs that are critically nonplanar include the Möbius ladder.

Critical nonplanar graphs are implemented in the Wolfram Language as GraphData["CriticalNonplanar"].

The numbers of critical nonplanar simple graphs on , 2, ... nodes are 0, 0, 0, 0, 1, 8, 40, 258, ... (OEIS A158922), the first few of which are illustrated above.

Apex Graph, Nonplanar Graph, Planar Graph, Triangulated Graph

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## References

Sloane, N. J. A. Sequence A158922 in "The On-Line Encyclopedia of Integer Sequences."Tucker, A. Applied Combinatorics, 4th ed. New York: Wiley, p. 43, 2001.

## Referenced on Wolfram|Alpha

Critical Nonplanar Graph

## Cite this as:

Weisstein, Eric W. "Critical Nonplanar Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CriticalNonplanarGraph.html