Is CMCG (Constrained Maximum-Weight Connected Graph) problem NP-complete?
Firstly, note that you can reduce $CMCG$ to $MCG$ by setting the value of the node that must be in the set to the  sum of all positive weights of all other nodes $+1$ and then applying $MCG$ and retrieving the result set. Thus $CMCG\in NP$.
As proved in the paper the $NP$-complete Steiner-Tree problem can be reduced to $CMCG$ meaning that $CMCG$ is $NP$-hard. If a problem is in $NP$ and $NP$-hard it is by definition $NP$-complete.
Hope this helps.