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

TL;DR

This paper develops structure theorems for graphs excluding certain topological minors, introduces the treebandwidth parameter, and provides algorithms for approximating and computing it efficiently.

Contribution

It introduces the concept of treebandwidth, extends bandwidth to tree layouts, and offers FPT algorithms for approximation and exact computation.

Findings

Structure theorems for graphs excluding fan or dipole minors.

FPT linear-time algorithms for approximating treebandwidth.

Complexity characterization for computing treebandwidth exactly.

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.