  Given an undirected graph $G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$,
balanced connected $q$-partition problem ($BCP_q$) asks whether there exists a
partition of the vertex set $V$ of $G$ into $q$ parts $V_1,V_2,V_3,\cdots, V_q$
such that for all $i\in[1,q]$, $|V_i|=n_i$ and the graph induced on $V_i$ is
connected. A related problem denoted as the balanced connected $q$-edge
partition problem ($BCEP_q$) is defined as follows. Given an undirected graph
$G$ and $q$ integers $n_1,n_2,n_3, \cdots, n_q$, $BCEP_q$ asks whether there
exists a partition of the edge set of $G$ into $q$ parts $E_1,E_2,E_3,\cdots,
E_q$ such that for all $i\in[1,q]$, $|E_i|=n_i$ and the graph induced on the
edge set $E_i$ is connected. Here we study both the problems for $q=2$ and
prove that $BCP_q$ for $q\geq 2$ is $W[1]$-hard. We also show that $BCP_2$ is
unlikely to have a polynomial kernel on the class of planar graphs. Coming to
the positive results, we show that $BCP_2$ is fixed parameter tractable (FPT)
parameterized by treewidth of the graph, which generalizes to FPT algorithm for
planar graphs. We design another FPT algorithm and a polynomial kernel on the
class of unit disk graphs parameterized by $\min(n_1,n_2)$. Finally, we prove
that unlike $BCP_2$, $BCEP_2$ is FPT parameterized by $\min(n_1,n_2)$.
