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I'm having problems finding the right words to formulate the following class of graphs in a definition. I'm defining a class of plane graphs with the following properties:

  • Removing any vertex of the outer face will not make the graph disconnected
  • Removing any two vertices not in the outer face will not make the graph disconnected.

The best definition I could find for now is to say $G$ is 2-connected on the outer face and 3-connected everywhere else.

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Can you give some counter examples? – draks ... Oct 9 '12 at 13:19

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