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I was going through some literature regarding Generalized Petersen graphs $P(n,k)$. Some authors considered $k < n$ and others took $k<\frac{n}{2}$. I am quite confused and don't have any idea which definition to pick for my work? Kindly help me in this regard. Thanks a lot for your kind help.

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It doesn't matter. If we allow $k > n/2$, we don't gain any generality, since $P(n,k)$ is the same graph as $P(n,n-k)$.

I guess that for even $n$, $P(n,n/2)$ is a graph only allowed by one of the definitions, but it's not a particularly interesting graph. Whether or not you want to allow it in the work you're doing presumably depends on what you're trying to do.

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