  Problems related to finding induced subgraphs satisfying given properties
form one of the most studied areas within graph algorithms. Such problems have
given rise to breakthrough results and led to development of new techniques
both within the traditional P vs NP dichotomy and within parameterized
complexity. The \Pi-Subgraph problem asks whether an input graph contains an
induced subgraph on at least k vertices satisfying graph property \Pi. For many
applications, it is desirable that the found subgraph has as few connections to
the rest of the graph as possible, which gives rise to the Secluded
\Pi-Subgraph problem. Here, input k is the size of the desired subgraph, and
input t is a limit on the number of neighbors this subgraph has in the rest of
the graph. This problem has been studied from a parameterized perspective, and
unfortunately it turns out to be W[1]-hard for many graph properties \Pi, even
when parameterized by k+t. We show that the situation changes when we are
looking for a connected induced subgraph satisfying \Pi. In particular, we show
that the Connected \Pi-Secluded Subgraph problem is FPT when parameterized by
just t for many important graph properties \Pi.
