Partitioning graphs of supply and demand Original Research Article

Assume that each vertex of a graph image is either a supply vertex or a demand vertex and is assigned a positive integer, called a supply or a demand. Each demand vertex can receive “power” from at most one supply vertex through edges in image. One thus wishes to partition image into connected components by deleting edges from image so that each component image has exactly one supply vertex whose supply is no less than the sum of demands of all demand vertices in image. If image does not have such a partition, one wishes to partition image into connected components so that each component image either has no supply vertex or has exactly one supply vertex whose supply is no less than the sum of demands in image, and wishes to maximize the sum of demands in all components with supply vertices. We deal with such a maximization problem, which is NP-hard even for trees and strongly NP-hard for general graphs. In this paper, we show that the problem can be solved in pseudo-polynomial time for series–parallel graphs and partial image-trees–that is, graphs with bounded tree-width.