Is CMCG (Constrained Maximum-Weight Connected Graph) problem NP-complete? MCG Problem: Consider a positive integer R and an undirected graph G = (V, E), in which each vertex is assigned a weight (or value). The maximum-weight connected graph (MCG) problem
is to find a subgraph with R vertices that is connected and maximizes the sum of the weights
(or values) of the R vertices.
CMCG Problem: The CMCG problem is the MCG problem with a constraint of having one predetermined (fixed) vertex included in the solution.
In 1 it has been stated (First paragraph, Section 1) that MCG is an NP-complete problem, which is proved in [3] (However I could not find the technical report [3] anywhere on the internet).
In 2 it has been &quot;claimed&quot; that CMCG is NP-complete (second to last paragraph, Section 3). But did not provide any proof.
I want to know (and see a proof, if possible) that if CMCG is NP-complete.
In 1 it has been shown that Steiner tree problem, which is NP-complete is a special case of the CMCG problem. Does it imply that CMCG is NP-complete?
[3]:Lee, H. F., and D. R. Dooly. The maximum-weight connected graph problem. Technical Report 93-4, Industrial Engineering, Southern Illinois University, 1993.
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.