Simple proof for NP-completeness of Edge Dominating Set
I assume that you only want to show the minimum edge dominating set problem is NP-complete for the general graph (i.e., you don't care about what properties the constructed graph has).

In case you missed it, there is an arguably simpler reduction from 3-SAT in the same paper (see Theorem 2). The reduction assumes no further "specialist" knowledge. You can make the presentation even slightly easier by skipping verification of bipartiteness and boundedness of the maximum degree of the resulting graph.