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I'm curious about some terminology for graphs and the existence of an algorithm. Let $G$ be a graph and $H \leq G$ a subgraph. Is there a name given to $H$ if $|N(H)|$ is minimum over all subgraphs of size $|H|$? Are there algorithms to compute such a set rather than brute force?

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If by $|N(H)|$ you mean neighborhood of $H$, then this is close to the notion of graph expansion.

As for your last question, I am positive, judging from inherent difficulty of calculating other expansion rates, I bet this one would be a hard problem as well. I am not sure, if there is nothing better than a brute force algorithm, but even if there is I wouldn't expect it to break the exponential barrier.

EDIT: Typos

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Thanks, that will give me something to work from. I don't have any votes left for today so if no one says anything else by the morning, I will +1 and accept this. –  muzzlator Apr 11 '13 at 19:02
    
Cheers mate! If you need anything more let me know. –  KXK Apr 11 '13 at 19:13
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