For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m^4 |L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivalent problems of enumerating minimal (vertex) dominating sets of line graphs and line graphs of bipartite graphs, with incremental-polynomial delay, and consequently in output-polynomial time. Enumeration of minimal dominating sets in graphs has very recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time is a fundamental and challenging question in Output-Sensitive Enumeration; it has been open for several decades and has triggered extensive research in the field.