  The classes Delay-FPT and Total-FPT recently have been introduced into
parameterized complexity in order to capture the notion of efficiently solvable
parameterized enumeration problems. In this paper we focus on ordered
enumeration and will show how to obtain Delay-FPT and Total-FPT enumeration
algorithms for several important problems. We propose a generic algorithmic
strategy, combining well-known principles stemming from both parameterized
algorithmics and enumeration, which shows that, under certain preconditions,
the existence of a so-called neighbourhood function among the solutions implies
the existence of a Delay-FPT algorithm which outputs all ordered solutions. In
many cases, the cornerstone to obtain such a neighbourhood function is a
Total-FPT algorithm that outputs all minimal solutions. This strategy is
formalized in the context of graph modification problems, and shown to be
applicable to numerous other kinds of problems.
