A cycle C of a graph G is called dominating cycle (D-cycle) if every edge of G is incident with at least one vertex of C. A D-path is defined analogously. If a graph G contains a D-cycle (D-path), then its edge graph L(G) has a hamiltonian cycle (hamiltonian path). Necessary conditions and sufficient conditions are obtained for graphs to have a D-cycle or D-path. They are analogous to known conditions for the existence of hamiltonian cycles or paths. The notions edge degree and remote edges arise as analogues of vertex degree and nonadjacent vertices, respectively. A result of Nash-Williams is improved.