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In graph theory, a connected dominating set and a maximum leaf spanning tree are two closely related structures defined on an undirected graph
In graph theory, a dominating set for a graph G (V, E) is a subset D of V such that every vertex not in D is adjacent to at least one member of D. The domination number γ(G) is the number of vertices in a smallest dominating set for G. The dominating set problem concerns testing whether γ(G) ≤ K for a given graph G and input K; it is a classical NP-complete decision problem in computational complexity theory.
First realize that a Vertex Cover of G is a Dominating Set of H (it's clearly a DS of G and the new vertices are also dominated). So if G has a VC smaller k, then H has a DS smaller k. For the converse, consider D, a Dominating Set of H. Notice that if one of the new vertices is in D, we can replace it with one of it's two neighbors and still get an Dominating Set: it's only neighbors are are the two original vertices and they are also connected - everything x can possible dominate is also dominated by u or v. So we can assume that D contains only vertices from G. Now for every. If I understood correctly your problem then it is not NP Complete.
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