Parameterized Complexity of Graph Partitioning into Connected Clusters

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 $\min(n_1,n_2)$.