Optimal Tree-Decompositions with Bags of Bounded Treewidth

2025-11-27

AI Summary

This paper addresses the problem of constructing optimal tree decompositions for several natural graph classes—including planar graphs, graphs of Euler genus $g$, and $K_p$-free graphs—with the dual objectives of minimizing treewidth and ensuring each bag has bounded treewidth. Employing techniques from tree decompositions, forbidden-subgraph theory, layered treewidth, and combinatorial topology, the authors establish, for the first time, that planar graphs admit treewidth-optimal tree decompositions where every bag has treewidth at most 3. This result extends to graphs of Euler genus $g$ and $K_p$-free graphs, yielding polynomial bounds $O(g)$ and $f(p)$, respectively. The work further demonstrates the necessity of forbidden-subgraph conditions for such guarantees and refutes the existence of additive-constant approximations for 1-planar graphs. Nevertheless, it provides an $O(sqrt{n})$-width tree decomposition for 1-planar graphs whose bags have bounded layered treewidth.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We prove that several natural graph classes have tree-decompositions with minimum width such that each bag has bounded treewidth. For example, every planar graph has a tree-decomposition with minimum width such that each bag has treewidth at most 3. This treewidth bound is best possible. More generally, every graph of Euler genus $g$ has a tree-decomposition with minimum width such that each bag has treewidth in $O(g)$. This treewidth bound is best possible. Most generally, every $K_p$-minor-free graph has a tree-decomposition with minimum width such that each bag has treewidth at most some polynomial function $f(p)$. In such results, the assumption of an excluded minor is justified, since we show that analogous results do not hold for the class of 1-planar graphs, which is one of the simplest non-minor-closed monotone classes. In fact, we show that 1-planar graphs do not have tree-decompositions with width within an additive constant of optimal, and with bags of bounded treewidth. On the other hand, we show that 1-planar $n$-vertex graphs have tree-decompositions with width $O(sqrt{n})$ (which is the asymptotically tight bound) and with bounded treewidth bags. Moreover, this result holds in the more general setting of bounded layered treewidth, where the union of a bounded number of bags has bounded treewidth.

Problem

• Proves optimal tree-decompositions with bounded bag treewidth for planar and minor-free graphs
• Shows analogous results fail for 1-planar graphs due to non-minor-closed properties
• Demonstrates 1-planar graphs have tree-decompositions with width O(sqrt(n)) and bounded bag treewidth

Innovation

• Tree-decompositions with minimum width and bounded bag treewidth
• Bounded treewidth bags for planar and minor-free graphs
• Width O(sqrt(n)) decompositions for 1-planar graphs with bounded bags