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

Takehiro Ito, Xiao Zhou, Takao Nishizeki
2006 Journal of Discrete Algorithms  
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)-partition. We deal with three problems to find an (l, u)-partition of a given graph; the minimum partition problem is to find an (l, u)-partition with the minimum number of
more » ... mponents; the maximum partition problem is defined analogously; and the p-partition problem is to find an (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(u 4 n) and the p-partition problem can be solved in time O(p 2 u 4 n) for any series-parallel graph with n vertices. The algorithms can be extended for partial k-trees, that is, graphs with bounded tree-width.
doi:10.1016/j.jda.2005.01.005 fatcat:oulsy2g5ofd67gv7ivjnktnjo4