Tree-partitions of graphs with given pathwidth
Provides a structural decomposition result for graphs with bounded pathwidth, relevant to graph theory and algorithm design.
The paper proves that graphs with bounded pathwidth and bounded maximum degree have tree-partitions of bounded width, with the underlying tree also having bounded pathwidth, and shows this bound is optimal up to a constant factor.
Graphs with bounded treewidth and bounded maximum degree are known to have tree-partitions of bounded width. What can be said if the bounded treewidth assumption is strengthened to bounded pathwidth? We prove that every graph with bounded pathwidth and bounded maximum degree has a tree-partition of bounded width, with the extra property that the underlying tree has bounded pathwidth. Moreover, we prove a lower bound showing that the bound on the pathwidth of the underlying tree is within a constant factor of optimal.