Planar Graph
A Graph is planar if it can be drawn in a Plane withoutEdges crossing (i.e., it has Crossing Number 0). Only planar graphshave Duals. If 
is planar, then has Vertex Degree 
. CompleteGraphs are planar only for 
. The complete Bipartite Graph 
in nonplanar. Moregenerally, Kuratowski proved in 1930 that a graph is planar Iff it does not contain within it any graph which can beContracted to the pentagonal graph 
or the hexagonal graph . 
can bedecomposed into a union of two planar graphs, giving it a ``Depth'' of 
. SimpleCriteria for determining the depth of graphs are not known. Beineke and Harary (1964, 1965) have shown that if
(mod 6), then
The Depths of the graphs
for 
, 10, 22, 28, 34, and 40 are 1, 3, 4, 5, 6, and 7 (Meyer1970).See also Complete Graph, Fabry Imbedding, Integral Drawing, Planar Straight Line Graph
References
Beineke, L. W. and Harary, F. ``On the Thickness of the Complete Graph.'' Bull. Amer. Math. Soc. 70, 618-620, 1964. Beineke, L. W. and Harary, F. ``The Thickness of the Complete Graph.'' Canad. J. Math. 17, 850-859, 1965. Booth, K. S. and Lueker, G. S. ``Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity using PQ-Tree Algorithms.'' J. Comput. System Sci. 13, 335-379, 1976. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 56, 1983. Meyer, J. ``L'épaisseur des graphes completes 
et 
.'' J. Comp. Th. 9, 1970.
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