$A$ set $F$ of edges in a graph $G (V, E)$ is a dominating edge set if every edge not in $F$ has a vertex in common with an edge in $F$. The edge domination number is the number of edges in a minimum edge domination set. Find the edge domination number of the graph of Fig. $1$-$14.$ ![Fig. 1-14]!](https://i.stack.imgur.com/NaaW7.png)
I've been thinking $\{7,4\}$ and $\{2,6\}$ is the minimum edge domination set, but $\{7,4\}$ and $\{2, 1\}$ is also possible as all the vertices are connected. I'm not sure how to think of this problem or if there's another way to think of it.