5.9.2 Characterizations of Planar Graphs

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• 5.9.2 Characterizations of Planar Graphs
• 1930
• Kuratowski (?????)
• Two basic nonplanar graphs K5 and K3,3

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• Definition 43 If a graph is planar, so will be
  any graph obtained by omitted an edge u,v and
  adding a new vertex w together with edges u,w
  and w,v. Such an operation is called an
  elementary subdivision.
• Definition 44 The graphs G1(V1,E1) and
  G2(V2,E2) are called homeomorphic if they can be
  obtained from the same graph by a sequence of
  elementary subdivisions.

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• Theorem 5.29 (1)If G has a subgraph homeomorphic
  to Kn, then there exists at least n vertices with
  the degree more than or equal n-1.
• (2) If G has a subgraph homeomorphic to Kn,n,
  then there exists at least 2n vertices with the
  degree more than or equal n.
• Example Let G(V,E),V7. If G has a subgraph
  homeomorphic to K5, then has not any
  subgraph homeomorphic to K3.3 or K5.

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• Theorem 5.30 Kuratowski