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Planar graph induction proof

  1. Nov 9, 2009 #1
    1. The problem statement, all variables and given/known data
    Let G be a planar graph with n vertices, q edges, and k connected components.
    If there are r regions in a planar representation of G, prove that:
    n − q + r = 1 + k
    Hint: Use induction on k. The base case is Euler’s formula.


    2. Relevant equations



    3. The attempt at a solution

    Alright,
    for k=1
    n-q+r=2 (this is true due to Euler's formula)

    now assume that this statement hold for k=p, meaning
    n-q+r=1+p

    This is where I get stuck, I don't know how to show it for k=p+1
    I am thinking it is something like since it is true for every one connected component, and p connected components that means that just adding one more to p would make the equation true as well. However, I don't know how to show that mathematically.... any help would be great!!
     
  2. jcsd
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