  One method of constructing $(a^2+1, 2,a, 1)$-SEDFs (i.e., strong external
difference families) in $\mathbb{Z}_{a^2+1}$ makes use of $\alpha$-valuations
of complete bipartite graphs $K_{a,a}$. We explore this approach and we provide
a classification theorem which shows that all such $\alpha$-valuations can be
constructed recursively via a sequence of ``blow-up'' operations. We also
enumerate all $(a^2+1, 2,a, 1)$-SEDFs in $\mathbb{Z}_{a^2+1}$ for $a \leq 14$
and we show that all these SEDFs are equivalent to $\alpha$-valuations via
affine transformations. Whether this holds for all $a > 14$ as well is an
interesting open problem. We also study SEDFs in dihedral groups, where we show
that two known constructions are equivalent.
