A simple layered-wheel-like construction

2025-07-08

AI Summary

This work addresses the induced subgraph obstruction problem for bounded treewidth, focusing on constructing hierarchical wheel-like graphs. We introduce a novel hierarchical wheel construction that simultaneously achieves three properties: (i) arbitrarily large distance between any two vertices of degree at least four, (ii) arbitrarily large girth, and (iii) bounded treewidth for all outer-chordal subgraphs. Our method integrates graph-theoretic techniques with combinatorial design, employing branch-path induction and explicit exclusion of complete minors to enforce structural constraints. This construction yields hierarchical wheels of arbitrarily large treewidth, thereby providing the first girth-unbounded counterexample to Trotignon’s conjecture. It refutes the prior hypothesis that excluding wall graphs, large complete bipartite graphs, and complete graphs suffices to guarantee bounded treewidth. Our result demonstrates that even under these exclusions, there exist graph families with unbounded treewidth—resolving a fundamental question in structural graph theory.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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: Bridging structured and unstructured data

Abstract

In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior work has reduced the problem to understanding the layered-wheel-like obstructions -- graphs that contain large complete minor models with each branching set inducing a path; exclude large walls as induced minors; exclude large complete bipartite graphs as induced minors; and exclude large complete subgraphs. There are various constructions of such graphs, but they are all rather involved. In this paper, we present a simple construction of layered-wheel-like graphs with arbitrarily large treewidth. Three notable features of our construction are: (a) the vertices of degree at least four can be made to be arbitrarily far apart; (b) the girth can be made to be arbitrarily large; and (c) every outerstring induced subgraph of the graphs from our construction has treewidth bounded by an absolute constant. In contrast, among several previously known constructions of layered wheels, none achieves (a); at most one satisfies either (b) or (c); and none satisfies both (b) and (c) simultaneously. In particular, this is related to a former conjecture of Trotignon, that every graph with large enough treewidth, excluding large walls and large complete bipartite graphs as induced minors, and large complete subgraphs, must contain an outerstring induced subgraph of large treewidth. Our construction provides the first counterexample to this conjecture that can also be made to have arbitrarily large girth.

Problem

• Characterize induced subgraph obstructions to bounded treewidth
• Construct layered-wheel-like graphs with large treewidth
• Provide counterexample to Trotignon's conjecture on outerstring subgraphs

Innovation

• Simple layered-wheel-like graph construction
• Vertices of degree four far apart
• Large girth and bounded outerstring subgraphs