Switching edges and vertices

Question

I am attempting to convert the problem of finding an edge dominating set into a dominating set. I need a way to change the edges of a graph to the vertices and the vertices of a graph to the edges, but I cannot seem to find one. Any help would be greatly appreciated.

Answer 1

You should look at the line graph of your original graph $G$. The line graph $L(G)$ takes vertices of $G$ and makes them edges. The edges of $G$ then become vertices.

So $E^{\prime} \subset E$ is an edge-dominating set if and only if $E^{\prime}$ is a dominating set for $L(G)$.