Tree-partitions of graphs with given pathwidth

2026-05-26

AI Summary

This work investigates whether graphs with bounded pathwidth and maximum degree admit a tree partition whose partition width and the pathwidth of its underlying tree are both bounded. By leveraging graph-theoretic analysis, pathwidth theory, and combinatorial optimization techniques, the study establishes—for the first time—the existence of such a tree partition for graphs under these constraints. Moreover, the derived upper bound on the pathwidth of the underlying tree is shown to be near-optimal up to a constant factor. These results collectively yield an existence theorem for tree partitions of bounded-pathwidth graphs and reveal favorable structural properties of such partitions under pathwidth constraints.

Abstract

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.

Problem

• tree-partition
• pathwidth
• bounded degree
• treewidth
• graph partitioning

Innovation

• tree-partition
• pathwidth
• bounded degree
• graph decomposition
• lower bound