Parameterized Complexity of Graph Partitioning into Connected Clusters

Given an undirected graph $G$ and $q$ integers $n1,n2,n3, \cdots, nq$, balanced connected $q$-partition problem ($BCPq$) asks whether there exists a partition of the vertex set $V$ of $G$ into $q$ parts $V1,V2,V3,\cdots, Vq$ such that for all $i\in[1,q]$, $|Vi|=ni$ 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 $n1,n2,n3, \cdots, nq$, $BCEPq$ asks whether there exists a partition of the edge set of $G$ into $q$ parts $E1,E2,E3,\cdots, Eq$ such that for all $i\in[1,q]$, $|Ei|=ni$ 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 $q\geq 2$ 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 $\min(n1,n2)$.