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We know that Hamiltonian path, clique and independent sets are NP-complete but what about half or the square root of each problem or a fraction of $n$ ? That is, for a graph, $G$ of $n$ nodes,

  • is there a Hamiltonian path of size $\frac{n}{2}$ or of $\sqrt{n}$ or any number of nodes less than $n$?
  • is there an independent set of size $b$ in $\frac{n}{2}$ or of $\sqrt{n}$ or any number of nodes less than $n$?
  • is there a clique of size $k$ in $\frac{n}{2}$ or of $\sqrt{n}$ or any number of nodes less than $n$?
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  • $\begingroup$ A hamiltonian path (if it exists) is necessarily of size $n$ in a graph of order $n$… $\endgroup$ – Nathaniel Apr 19 at 16:25
  • $\begingroup$ I don't understand the problem statements. For example, what's the intended difference between the second problem and "is there an independent set of size b on a given graph"? $\endgroup$ – Juho Apr 19 at 18:52
  • $\begingroup$ Instead of an independent set with $G$ of size $n$, finding an independent set of a subgraph, $G'$, of $\frac{n}{2}$ nodes that are from $G$ $\endgroup$ – heretoinfinity Apr 19 at 20:36
  • $\begingroup$ It looks like you're asking about 6 different questions here. There is no reason that the answer will necessarily be the same for all of them. The usual rule is to ask only one question per post $\endgroup$ – D.W. Apr 20 at 2:34
  • $\begingroup$ @D.W., do you mean that each of the bullet points is actually a set of 2 questions? $\endgroup$ – heretoinfinity Apr 20 at 14:53
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There is no general rule that they necessarily need to be NP-complete in general.

For the specific examples you list, I expect the answer will be that each of those specific examples are NP-complete, but I haven't tried to prove or verify that; that's just speculation.

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