Understanding Reductions to show NP-Completeness

Question

I have a homework problem that I am finding difficult to begin. We are working on Karp (single-call) reductions to show intractability. For this assignment, the problem is intentionally vague. I was hoping someone here may know it or can provide an example of a solution to help me get started.

The problem is described only in code they provided. It has the following components:

B = (G, T, k) where B is an instance of the "Blah" problem, G = (V,E) is a graph (unweighted, undirected), T is a subset of vertices in V, and k is an integer. The certificate for B returns a subgraph, S = (V', E'), of G. The verifier for a yes instance is also provided:

a yes instance if

forall [t in T], there exists some edge [e in E'] s.t. t is an endpoint of e
and, the number of edges of S is at most k (|E'| <= k)
and, S is a connected graph

I see some similarities between this problem and the Independent-Set or Vertex-Cover problems. I feel these are good candidates to reduce from to show intractability, but do not yet understand the problem well enough. If there is discussion of this problem somewhere, or anyone can provide some examples, I would greatly appreciate it. Thank you!