  There is substantial literature dealing with fixed parameter algorithms for
the dominating set problem on various families of graphs. In this paper, we
give a $k^{O(dk)} n$ time algorithm for finding a dominating set of size at
most $k$ in a $d$-degenerated graph with $n$ vertices. This proves that the
dominating set problem is fixed-parameter tractable for degenerated graphs. For
graphs that do not contain $K_h$ as a topological minor, we give an improved
algorithm for the problem with running time $(O(h))^{hk} n$. For graphs which
are $K_h$-minor-free, the running time is further reduced to $(O(\log
h))^{hk/2} n$. Fixed-parameter tractable algorithms that are linear in the
number of vertices of the graph were previously known only for planar graphs.
  For the families of graphs discussed above, the problem of finding an induced
cycle of a given length is also addressed. For every fixed $H$ and $k$, we show
that if an $H$-minor-free graph $G$ with $n$ vertices contains an induced cycle
of size $k$, then such a cycle can be found in O(n) expected time as well as in
$O(n \log n)$ worst-case time. Some results are stated concerning the
(im)possibility of establishing linear time algorithms for the more general
family of degenerated graphs.
