Given an undirected graph G and q integers n1, n2, n3, • • • , nq, balanced connected qpartition problem (BCPq) asks whether there exists a partition of the vertex set V of G into q parts V1, V2, V3, • • • , Vq such that for all i ∈ [1, q], |Vi| = ni and the graph induced on Vi is connected. A related problem denoted as the balanced connected q-edge partition problem (BCEPq) is defined as follows. Given an undirected graph G and q integers n1, n2, n3, • • • , nq, BCEPq asks whether there exists a partition of the edge set of G into q parts E1, E2, E3, • • • , Eq such that for all i ∈ [1, q], |Ei| = ni and the graph induced on the edge set Ei is connected.Here we study both the problems for q = 2 and prove that BCPq for q ≥ 2 is W [1]-hard. We also show that BCP2 is unlikely to have a polynomial kernel on the class of planar graphs.Coming to the positive results, we show that BCP2 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(n1, n2). Finally, we prove that unlike BCP2, BCEP2 is FPT parameterized by min(n1, n2).