Partitioning a graph of bounded tree-width to connected subgraphs of almost uniform size

Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wishes to partition G into connected components by deleting edges from G so that the total weight of each component is at least l and at most u  . Such an “almost uniform” partition is called an (l,u)(l,u)-partition. We deal with three problems to find an (l,u)(l,u)-partition of a given graph; the minimum partition problem is to find an (l,u)(l,u)-partition with the minimum number of components; the maximum partition problem is defined analogously; and the p  -partition problem is to find an (l,u)(l,u)-partition with a fixed number p   of components. All these problems are NP-complete or NP-hard, respectively, even for series-parallel graphs. In this paper we show that both the minimum partition problem and the maximum partition problem can be solved in time O(u4n)O(u4n) and the p  -partition problem can be solved in time O(p2u4n)O(p2u4n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.