Polynomial algorithms for graph isomorphism and chromatic index on partial k-trees
Journal of Algorithms
Easy problems for tree-decomposable graphs
Journal of Algorithms
Most uniform path partitioning and its use in image processing
Discrete Applied Mathematics - Special issue: combinatorial structures and algorithms
Journal of the ACM (JACM)
Linear-time computability of combinatorial problems on series-parallel graphs
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Operating systems
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Assume that each vertex of a graph G is assigned a nonnegative integer weight and that l and u are nonnegative integers. One wish 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 $(l, u) \mbox{-}$partition. We deal with three problems to find an $(l, u) \mbox{-}$partition of a given graph. The minimum partition problem is to find an $(l, u) \mbox{-}$partition with the minimum number of components. The maximum partition problem is defined similarly. The p-partition problem is to find an $(l, u) \mbox{-}$partition with a fixed number p of components. All these problems are NP-complete or NP-hard 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) and the p-partition problem can be solved in time O(p2u4n) for any series-parallel graph of n vertices. The algorithms can be easily extended for partial k-trees, that is, graphs with bounded tree-width.