It's well-known that, for any $n$, we can consider
$D_n = \langle r, f | r^n = e, f^2 = e, r^k \not = e (0 < k < n), fr= r^{-1} f\rangle$;
However, not all two element generating sets for $D_n$ satisfy these conditions: if we take $\{f, fr\}$. then $D_n = \langle f, fr \rangle$, but these two elements have order two. So my questions is: are there general (and interesting) conditions such that any two element generating set for $D_n$ must satisfy?
The characterization for a generating pair $x,y$ is either one of $x,y$ (say $x$) is a reflection and $y$ generates the group of rotations (so its order is $n$), or $x,y$ are both reflections and the order of $xy$ is $n$. It's easy to go from one presentation to the other; given a generating set of the second type, $\{x,xy\}$ is a generating set of the first type, and given a generating set of the first type, $\{x,xy\}$ is a generating set of the second type. It's not too hard to see that this exhausts all possibilities. Try to prove it.