I know the method to the find the minimum pumping length of regular language by constructing minimal DFA and finding the number of states but I am not able to quite understand why is it working.
For example, $L = (10)^*$. Here minimum pumping length is 1 by constructing DFA. But by pumping lemma, how can there be a string of length 1 belonging to $L$? As pumping lemma says the string $S$ should be such that $|S| \le p$ and $S$ should belong to $L$, where $p$ is the pumping length, and here there is no string of length 1 belonging to $L$.