On the dominating (induced) cycles of iterated line graphs

In this paper, we give characterizations of graphs with line graphs or iterated line graphs that have dominating cycles. The characterization of graphs with dominating cycles in its line graphs and its i-iterated line graphs for i≥2 are different: we may not unify them. As an application, we give characterizations of graphs with iterated line graphs that have dominating induced cycles. They are very different from the known results, although those characterizations for dominating cycles have some similarities with results on hamiltonian iterated line graphs of Harary and Nash-Williams (1965) and Xiong and Liu (2002). Using these results, we also give some analysis on the complexity of determining the existence of dominating cycles. It is NP-complete to decide whether a given graph has a dominating induced cycle, even for a 2-iterated line graph.