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| Branchwidth | |
|---|---|
| Name | Branchwidth |
| Field | Graph theory, Matroid theory |
| Related | Treewidth, Rankwidth, Branch decomposition, Tutte polynomial |
Branchwidth Branchwidth is an invariant of graphs and matroids that quantifies how a structure can be decomposed along small edge cuts; it plays a central role in structural graph theory and algorithmic graph minor theory. It is closely connected to other width parameters such as Treewidth and Rankwidth and underlies fixed-parameter tractable algorithms for many combinatorial problems studied by researchers at institutions such as University of Oxford, California Institute of Technology, and research groups around the Graph Minors project. The concept appears in literature alongside key results like the Robertson–Seymour theorem and tools such as the Tutte polynomial.
Branchwidth is defined via a branch decomposition: a subcubic tree whose leaves correspond to elements (edges of a graph or elements of a matroid). For a given edge of the decomposition, the width is the size of a minimal separating set (sometimes measured as the number of vertices incident to both sides for graphs or the rank of a partition for matroids). The branchwidth of the object is the minimum, over all decompositions, of the maximum width encountered. Important properties include monotonicity under taking minors (used in the Graph Minors series), relationships with connectivity concepts such as k-connectivity and bounds comparing branchwidth to other parameters like Pathwidth and Treewidth.
Branchwidth and Treewidth are tightly linked by inequalities: for any graph G, branchwidth(G) and treewidth(G) are within constant factors of each other, with classical bounds established by Robertson–Seymour and refined in subsequent work by researchers from institutions including Princeton University and ETH Zurich. Rankwidth is another width parameter introduced by researchers at Université Paris-Sud and Nihon University that behaves differently on dense graphs; comparisons between rankwidth and branchwidth appear in studies by authors affiliated with Carnegie Mellon University and Universität Bonn. These relationships are exploited in algorithm design for problems previously studied by investigators at places such as Massachusetts Institute of Technology and University of Cambridge.
A branch decomposition is a labeled tree structure used to guide dynamic programming; its development was advanced in the Graph Minors project and by algorithmic graph theorists at Bell Labs and Bellairs Research Institute. Algorithms that use branch decompositions perform divide-and-conquer along the decomposition tree and compute local information at each internal node, an approach used by teams at University of Waterloo and University of Illinois Urbana-Champaign to obtain fixed-parameter tractable routines for problems like Hamiltonian cycle and Graph coloring under bounded width constraints. Implementations often leverage reductions by groups such as those at Google Research and optimizations inspired by classical work at AT&T Labs.
Computing exact branchwidth is NP-hard in general, a complexity result proved by reductions similar to those used in hardness proofs by researchers linked to Stanford University and Columbia University. For fixed k, recognition of graphs of branchwidth at most k is solvable in polynomial time via algorithms derived from the Robertson–Seymour framework and later made more practical by work from University of Edinburgh and University of Sydney. Approximation algorithms and parameterized algorithms for branchwidth exploit connections to Treewidth and use techniques developed at Microsoft Research and Max Planck Institute for Informatics.
Branchwidth is applied to structural decompositions in studies of graph minors led by teams at Imperial College London and the University of Toronto, to obtain powerful structural theorems used in proofs of results like the Graph Minor Theorem. In Matroid theory, branchwidth provides a natural measure for decomposability and is used in algorithms for matroid optimization problems studied by researchers at Cornell University and University of California, Berkeley. The parameter also arises in work on the Tutte polynomial, network reliability analyses pursued at Duke University, and in the study of topological graph embeddings investigated by scholars at University of Amsterdam.
Classical families illustrate the range of branchwidth values: trees have small branchwidth (related to results from Harvard University combinatorial seminars), grids yield high branchwidth and serve as obstructions in the Graph Minors theory developed by University of California, San Diego affiliates, and complete graphs achieve large branchwidth proportional to their size, a fact noted in combinatorics literature from Rutgers University. Special cases include series–parallel graphs (bounded branchwidth, studied at Brown University), cographs (connections to Rankwidth examined at University of British Columbia), and planar graphs where branchwidth is tightly related to planar separators investigated by groups at INRIA and University of Tokyo.