Partitioning the Edges of a Planar Graph into Two Partial K-Trees

In this paper we prove two results on partitioning the edges of a planar graph into two partial k-trees, for fixed values of k. Interest in this class of partitioning problems arises since many intractable graph and network problems admit polynomial time solutions on k-trees and their subgraphs (partial k-trees). The first result shows that every planar graph is a union of two partial 3-trees. Furthermore, such a partitioning can be computed in linear time. Second, we show a recursive procedure to construct an infinite family of planar graphs in which every member does not admit a partitioning into a partial 1-tree (forest) and a partial 2-tree (series-parallel graph).

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