  Given a simple connected undirected graph G = (V, E), a set X \subseteq V(G),
and integers k and p, STEINER SUBGRAPH EXTENSION problem asks if there exists a
set S \supseteq X with at most k vertices such that G[S] is p-edge-connected.
This is a natural generalization of a well-studied problem STEINER TREE (set
p=1 and X as the set of all terminals). In this paper, we initiate the study of
STEINER SUBGRAPH EXTENSION from the perspective of parameterized complexity and
give a fixed-parameter algorithm parameterized by k and p on graphs of bounded
degeneracy. In case we remove the assumption of the input graph being bounded
degenerate, then the STEINER SUBGRAPH EXTENSION problem becomes W[1]-hard.
Besides being an independent advance on the parameterized complexity of network
design problems, our result has natural applications. In particular, we use our
result to obtain singly exponential-time FPT algorithms for several vertex
deletion problem studied in the literature, where the goal is to delete a
smallest set of vertices such that (i) the resulting graph belongs to a
specific hereditary graph class, and (ii) the deleted set of vertices induces a
p-edge-connected subgraph of the input graph.
