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Hamiltonian path & cycle

  1. Oct 10, 2011 #1
    Watching this lecture from nptel about NP completeness:
    http://www.youtube.com/watch?v=76n4BjlL1cs&feature=player_embedded

    -We have an algorithm, HC , which given a graph tells us whether or not it has a hamiltonian cycle.
    -We want to use it in order to create an algorithm that determines whether an input graph has a hamiltonian path.


    If the input has a HC, then it has a HP as well.Okay.

    But around 29:00, it is stated:

    If the input doesn't have a HC (the output of HC algorithm is a "no"), we add a vertex to it that connects to all other vertices in the graph.If that new gragh G' has a HC, then the original graph G has a HP.
    But thinking about this, how bout this example?

    edit:in the pic, i meant to say that "graph G doesn't have a HP"
     

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    Last edited: Oct 10, 2011
  2. jcsd
  3. Oct 14, 2011 #2
    You've misunderstood what the goal is. The goal is to find out whether G has a HP.
    See: http://www.youtube.com/watch?v=76n4BjlL1cs&feature=youtu.be#t=21m08s"

    The input is the Graph G' which consists of the original Graph G and the additional vertex u connected to all vertices of G.

    We want to find out if G has a HP by using the following theorem:
    G has a HP iff G' has a HC

    In your example, let us pretend that we do not know whether G has a HP.
    So, we form G' and use it as an input for HC. Now, there are two possible outcomes:
    (i) HC says: yes, G' has a HC. It then follows that G has an HP.
    (ii) HC says: no, G' has no HC. It then follows that G has no HP.
     
    Last edited by a moderator: Apr 26, 2017
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