Doubly Balanced Connected Graph Partitioning
  We introduce and study the Doubly Balanced Connected graph Partitioning (DBCP) problem: Let $G=(V,E)$ be a connected graph with a weight (supply/demand) function $p:V\rightarrow \{-1,+1\}$ satisfying $p(V)=\sum_{j\in V} p(j)=0$. The objective is to partition $G$ into $(V_1,V_2)$ such that $G[V_1]$ and $G[V_2]$ are connected, $|p(V_1)|,|p(V_2)|\leq c_p$, and $\max\{\frac{|V_1|}{|V_2|},\frac{|V_2|}{|V_1|}\}\leq c_s$, for some constants $c_p$ and $c_s$. When $G$ is 2-connected, we show that a solution with $c_p=1$ and $c_s=3$ always exists and can be found in polynomial time. Moreover, when $G$ is 3-connected, we show that there is always a `perfect' solution (a partition with $p(V_1)=p(V_2)=0$ and $|V_1|=|V_2|$, if $|V|\equiv 0 (\mathrm{mod}~4)$), and it can be found in polynomial time. Our techniques can be extended, with similar results, to the case in which the weights are arbitrary (not necessarily $\pm 1$), and to the case that $p(V)\neq 0$ and the excess supply/demand should be split evenly. They also apply to the problem of partitioning a graph with two types of nodes into two large connected subgraphs that preserve approximately the proportion of the two types. 
Saleh Soltan, Mihalis Yannakakis, and Gil Zussman
math.CO cs.CC cs.DM cs.DS