Is it NP hard to find a minimum dominating set containing some desired vertices?

Question

For a connected undirected graph, G = (V, E)

And a desired vertex set D, D is a subset of V (i.e. D \in V)

Is it NP-hard to find a minimum dominating set, containing the desired vertex set D?

Answer 1

Yes, it is a NP-hard problem. Please refer to the following document to read the reduction. Feel free to ask if you have problems in understanding the proof.

http://www.cs.iastate.edu/~chaudhur/cs611/Sp07/notes/lec22.pdf

To explain a bit more on your problem, i.e. adding the restriction D is a subset of V.....think like this -- when you are trying to prove your problem is NP, you reduce a known NP problem to a specific instance of your problem. Your specific instance of the problem can be a case when D=V...and you can prove your problem is also NP. Hope this helps.