Partition a graph into 2 connected subgraphs
The problem you asked is the unweighted version of the Balance Connected 2-Partition ($BCP_2$).

For unweighted case, any 2-connected graph can be partitioned into two connected subgraphs whose numbers of vertices differ by at most one. A simple algorithm uses st-numbering. For any 2-connected graph, we can label the vertices by $[1...n]$ such that any vertex has simultaneously a neighbor with smaller label and a neighbor with larger label. Let $V_1=\{1...n/2\}$ and $V_2=V-V_1$. It can be easily shown that both $V_1$ and $V_2$ induce connected subgraphs.

However, when there are cut vertices, the problem is NP-hard because it is equivalent to the weighted $BCP_2$. The transformation is as follows. Let $v$ be a cut vertex and $H$ be the maximum connected component in $G-v$. We shrink all components other than $H$ into $v$ and the weight of $v$ is given by the weight of the vertices combined in $v$. Repeat this process and we can obtain a weighted 2-connected graph.

It can be easily realized that there exists a graph such that the minimum part contains only $n/3$ vertices in any 2-partition.

For $BCP_2$, the currently best approximation algorithm is due to Chlebikova (I hope that it is not out of date): 

Approximating the maximally balanced connected partition problem in graphs, Information Processing Letters, 60:225--230, 1996. The approximation ratio is 4/3. For some special graphs, there are better results. For example, FPTAS for interval graphs and 5/4-approximation for grid graphs (further improved to 7/6).