Let \(G=(V,E)\) be a finite graph. The cardinality of \(V\) (the set of vertices) is \(n\) and the cardinality of \(E\) (the set of edges) is \(m\). Define the following relation of \(E\): \(e\succeq e'\) iff either \(e=e'\) or \(e\) and \(e'\) are adjacent. A subset \(D\subseteq E\) is called an edge dominating set if for each \(e'\in E\) there is \(e\in D\) such that \(e\succeq e'\). Let \(\gamma'(G)\) be the cardinality of a smallest edge dominating set. A subset \(I\subseteq V\) is a stable set if no two of its vertices are adjacent. Let \(\alpha(G)\) be the cardinality of a largest stable set of \(G\). A subset \(T\subseteq V\) is a 2-stable set if the distance between any two vertices in \(T\) is greater than 2. Let \(\alpha_ 2(G)\) be the cardinality of a largest 2-stable set of \(G\). The subdivision graph of \(G\) is \(S(G)=(V\cup E,E')\) where \(E'=\{\{e,v\}\): \(e\in E\), and \(v\in V\) is incident with \(e\}\). Finally, the total graph of \(G\) is defined by \(T(G)= \{V\cup E,\;E\cup E'\cup E^*\}\) where \(E^*=\{\{e,f\}\): \(e\) and \(f\) are adjacent edges of \(E\}\). The following results are proved for any \(G\): \(\gamma'(S(G))+ \alpha_ 2(G)= n\), \(2\gamma'(T(G))+ \alpha(T(G))= n+m\) or \(n+m+1\). Also, for any depth-first search tree \(S\) of \(G\), \({\gamma'(S)\over 2}\leq \gamma'(G)\leq 2\gamma'(S)\) and these bounds are tight. The complexity of the edge domination problem is also analyzed.