Non-planar graph

Figure 33. Any non-planar graph contains one of these graphs.

Intrinsically non-planar graphs are non-planar by virtue of their connectivity only. It has been shown [51] that they all contain one of the two basic non-planar graphs as a subgraph [60] the first and the second Kuratowski graphs K5 and... 

Extrinsically non-planar graphs are non-planar by virtue of their embedding in the 3D-space. They may be intrinsically planar at the same time (Fig. 6) this is the case of links (45) which are homeomorphic to disjoint sets of circuit graphs [71,72] (46) and knots (47a and b) which are homeomorphic to a circuit graph (48). 

At the time they were synthesized, these molecules were not recognized as having a non-planar graph, and actually the first topologically chiral K5 molecule recognized as such, is the centered polyquinane derivative 53, independently synthesized by H. E. Simmons and J. E. Maggio [80] on the one hand, and by L. A. Paquette and M. Vazeux [81] on the other hand (Fig. 9). That it is topologi-... 

Figure 42. The smallest non-planar graphs of Kura-towski 02 the complete graph on five vertices K, and the complete bipartite graph K3 3 having six vertices.

Definition Non-planar graphs cannot be drawn without at least one crossing of edges. 

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