  We investigate the parameterized complexity of finding subgraphs with
hereditary properties on graphs belonging to a hereditary graph class. Given a
graph $G$, a non-trivial hereditary property $\Pi$ and an integer parameter
$k$, the general problem $P(G,\Pi,k)$ asks whether there exists $k$ vertices of
$G$ that induce a subgraph satisfying property $\Pi$. This problem,
$P(G,\Pi,k)$ has been proved to be NP-complete by Lewis and Yannakakis. The
parameterized complexity of this problem is shown to be W[1]-complete by Khot
and Raman, if $\Pi$ includes all trivial graphs but not all complete graphs and
vice versa; and is fixed-parameter tractable (FPT), otherwise. As the problem
is W[1]-complete on general graphs when $\Pi$ includes all trivial graphs but
not all complete graphs and vice versa, it is natural to further investigate
the problem on restricted graph classes.
  Motivated by this line of research, we study the problem on graphs which also
belong to a hereditary graph class and establish a framework which settles the
parameterized complexity of the problem for various hereditary graph classes.
In particular, we show that:
  $P(G,\Pi,k)$ is solvable in polynomial time when the graph $G$ is
co-bipartite and $\Pi$ is the property of being planar, bipartite or
triangle-free (or vice-versa).
  $P(G,\Pi,k)$ is FPT when the graph $G$ is planar, bipartite or triangle-free
and $\Pi$ is the property of being planar, bipartite or triangle-free, or graph
$G$ is co-bipartite and $\Pi$ is the property of being co-bipartite.
  $P(G,\Pi,k)$ is W[1]-complete when the graph $G$ is $C_4$-free,
$K_{1,4}$-free or a unit disk graph and $\Pi$ is the property of being either
planar or bipartite.
