R-BBG$_2$: Recursive Bipartition of Bi-connected Graphs

Given an undirected graph $G(V, E)$, it is well known that partitioning a graph $G$ into $q$ connected subgraphs of equal or specificed sizes is in general NP-hard problem. On the other hand, it has been shown that the q-partition problem is solvable in polynomial time for q-connected graphs. For example, efficient polynomial time algorithms for finding 2-partition (bipartition) or 3-partition of 2-connected or 3-connected have been developed in the literature. In this paper, we are interested in the following problem: given a bi-connected graph $G$ of size $n$, can we partition it into two (connected) sub-graphs, $G[V_1]$ and $G[V_2]$ of sizes $n_1$ and $n_2$ such as both $G[V_1]$ and $G[V_2]$ are also bi-connected (and $n_1+n_2=n$)? We refer to this problem as the recursive bipartition problem of bi-connected graphs, denoted by R-BBG$_2$. We show that a ploynomial algorithm exists to both decide the recursive bipartion problem R-BBG$_2$ and find the corresponding bi-connected subgraphs when such a recursive bipartition exists.