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Partitioning Planar Graphs with Vertex Costs: Algorithms and Applications

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Abstract.

We prove separator theorems in which the size of the separator is minimized with respect to non-negative vertex costs. We show that for any planar graph G there exists a vertex separator of total sum of vertex costs at most \(c\sqrt{\sum_{v\in V(G)}( cost (v))^2}\) and that this bound is optimal to within a constant factor. Moreover, such a separator can be found in linear time. This theorem implies a variety of other separation results. We describe applications of our separator theorems to graph embedding problems, to graph pebbling, and to multicommodity flow problems.

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Received June 1997; revised February 1999.

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Djidjev, H. Partitioning Planar Graphs with Vertex Costs: Algorithms and Applications . Algorithmica 28, 51–75 (2000). https://doi.org/10.1007/s004530010031

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  • Key words. Graph separators, Divide and conquer, Graph embeddings, Pebbling, Multicommodity flow.