I was reading the paper "Kou, L. T., Stockmeyer, L. J., & Wong, C. K. (1978). Covering edges by cliques with regard to keyword conflicts and intersection graphs. Communications of the ACM, 21(2), 135-139" and from what I understand, they prove that SET-ECC is NP-complete by constructing a graph G' and showing that (G, k)$\in$ SET-NCC iff (G', k') $\in$ SET-ECC, with $k'= k(e + 1) + e$. However, the size of the ECC of G' could be smaller than k', and in fact, we don't know exactly what it is. I just want to confirm that I understood the paper correctly, so my question is, for a reduction of this type, is it enough to have an upper bound of k', as long as k' depends on k?
A mapping $f$ from instances of a decision problem $A$ to instances of a decision problem $B$ is a reduction if $x \in A$ iff $f(x) \in B$.
In your case, instances of both problems are of the form $(G,k)$. Such instances are Yes instances if there is a clique-cover of $G$ of size at most $k$.
Thus a function mapping $(G,k)$ to $(G',k')$ is a reduction between SET-NCC and SET-ECC if the following holds: $G$ has a node-clique-cover of size at most $k$ iff $G'$ has an edge-clique-cover of size at most $k'$.