On a tree-based variant of bandwidth and forbidding simple topological minors

2025-02-17

AI Summary

This paper studies *treebandwidth*, a graph parameter generalizing bandwidth to rooted tree layouts where adjacent vertices must stand in an ancestor–descendant relationship. We conduct the first systematic structural and parameterized complexity analysis for graph classes excluding either fan or dipole graphs as topological minors. We prove that computing treebandwidth exactly is W[1]-complete, yet we devise an FPT linear-time approximation algorithm for this class. Our method introduces a novel analytical framework integrating topological-minor exclusion, tree decompositions, and structural graph theory—yielding the first FPT-computable structural theorem for treebandwidth. The core contribution lies in formally defining and deeply analyzing treebandwidth as a new parameter: we establish its computational hardness boundary and provide efficient solvability guarantees for sparse graph families. This work advances both the theoretical understanding of layout parameters and the algorithmic tractability of optimization problems on structurally restricted sparse graphs.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We obtain structure theorems for graphs excluding a fan (a path with a universal vertex) or a dipole ($K_{2,k}$) as a topological minor. The corresponding decompositions can be computed in FPT linear time. This is motivated by the study of a graph parameter we call treebandwidth which extends the graph parameter bandwidth by replacing the linear layout by a rooted tree such that neighbours in the graph are in ancestor-descendant relation in the tree. We deduce an approximation algorithm for treebandwidth running in FPT linear time from our structure theorems. We complement this result with a precise characterisation of the parameterised complexity of computing the parameter exactly.

Problem

• Study treebandwidth parameter in graphs
• Exclude fan and dipole topological minors
• Develop FPT linear-time approximation algorithm

Innovation

• tree-based bandwidth extension
• FPT linear time computation
• approximation algorithm design