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We introduce and study the doubly balanced connected graph partitioning problem: Let G =( V , E ) be a connected graph with a weight (supply/demand) function p : V → {−1, +1} satisfying p ( V )=∑ j &isin 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 )∣≤ c p , and max{ ∣ V 1 / V 2 ∣,∣ V 2 / V 1 ∣} ≤ 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 =2 always exists and can be found in randomized 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 ∣≡ 0 (mod 4)), and it can be found in randomized polynomial time. Our techniques can be extended, with similar results, to the case in which the weights are arbitrary (not necessarily ±1), and to the case that p ( V )≠ 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.