Max-weight connected & co-connected subgraph problem
Problem: 
  Given a graph $G=(V,E)$ and a weight function $w:V\to\mathbb{R}$, find 
  a partition of $V$ into two parts $V_1$ and $V_2$, so that $G[V_1]$ and $G[V_2]$ are connected and so that $w(V_1)$ is maximized.


This problem is NP-hard, as follows easily from the following paper:


  Pim van 't Hof, Daniël Paulusma, Gerhard J. Woeginger:
  Partitioning graphs into connected parts. Theor. Comput. Sci. 410(47-49): 4834-4843 (2009)


The paper shows that the following problem is NP-hard: Given a graph $G=(V,E)$ and two vertices $s,t\in V$, is there a "good" partition of $V$ into two sets $V_1$ and $V_2$, so that $G[V_1]$ and $G[V_2]$ are connected and so that $s,t\in V_1$. 

Take such an instance, and make $w(s)=w(t)=1$ and $w(x)=0$ for $x\in V-\{s,t\}$. If there exists a good partition, you can reach $w(V_1)=2$.
If there is no good parition, the best you can get is $w(V_1)=1$.
(This argument shows NP-hardness, and also in-approximability within a
factor of $2-\epsilon$.)