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Branch-decomposition: Art, Forbidden minors & Algorithms and complexity

In graph theory, a branch-decomposition of an undirected graph G is a hierarchical clustering of the edges of G, represented by an unrooted binary tree T with the edges of G as its leaves. Removing any edge from T partitions the edges of G into two subgraphs, and the width of the decomposition is the maximum number of shared vertices of any pair of…

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Branch-decomposition topic overview

The analysis highlights Art, Forbidden minors and Algorithms and complexity as prominent areas in the source structure around Branch-decomposition.

Related topics
50
Source areas
7
Connected nodes
57
Extracted relationships
32
Concept neighborhoods
32
Bridge connections
57

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Forbidden minors · 13 topics
Algorithms and complexity · 12 topics
Generalization to matroids · 11 topics
Overview · 9 topics
Relation to treewidth · 3 topics
Carving width · 1 topics
Definitions · 1 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Definitions

Relation to treewidth

Carving width

Algorithms and complexity

Generalization to matroids

Forbidden minors

Advanced semantic analysis

Deeper signals for content research, entity SEO and topical coverage. The plain-language headings explain what each technical view is useful for.

How Branch-decomposition connects Entity context

The extracted context around Branch-decomposition shows recurring relationship patterns in the source. For example, Branch-decomposition → An, E1, E2, G1, G2, If, T1, T2, The, This Another extracted example is Branch-decomposition → Additionally, An, Branchwidth, However, If, It, Robertson, Seymour, The. Use these groups to spot repeated connection types before inspecting the individual relationships.

Branch-decomposition

Top relations

related to Definitions · 10
Branch-decomposition → An, E1, E2, G1, G2, If, T1, T2, The, This
related to Generalization to matroids · 9
Branch-decomposition → Additionally, An, Branchwidth, However, If, It, Robertson, Seymour, The
related to Algorithms and complexity · 8
Branch-decomposition → For, However, It, NP-complete, Paul Seymour, Robin Thomas, The, This
related to Relation to treewidth · 4
Branch-decomposition → Branch-decompositions, In, Neil Robertson, Paul Seymour
is a · 1
Branch-decomposition → maximum width of any of its e-separations

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

branchwidth graph graphs width treewidth forbidden matroid may matroids minors tree two planar time edges minimal seymour unrooted binary decomposition

Branch-decomposition relationships Subject–Predicate–Object triples

TTTA extracted 32 structured relationships around Branch-decomposition. Examples in this analysis include Branch-decomposition → is a → maximum width of any of its e-separations and Branch-decomposition → related to Algorithms and complexity → It. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Branch-decompositionis amaximum width of any of its e-separations0.90text
Branch-decompositionrelated to Algorithms and complexityIt0.60section
Branch-decompositionrelated to Algorithms and complexityNP-complete0.60section
Branch-decompositionrelated to Algorithms and complexityHowever0.60section
Branch-decompositionrelated to Algorithms and complexityFor0.60section
Branch-decompositionrelated to Algorithms and complexityThis0.60section
Branch-decompositionrelated to Algorithms and complexityThe0.60section
Branch-decompositionrelated to Algorithms and complexityPaul Seymour0.60section
Branch-decompositionrelated to Algorithms and complexityRobin Thomas0.60section
Branch-decompositionrelated to DefinitionsAn0.60section
Branch-decompositionrelated to DefinitionsIf0.60section
Branch-decompositionrelated to DefinitionsT10.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Branch-decomposition bring nearby vocabulary together. In this analysis, examples include Leaves, Binary and Unrooted. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • graph theory
    • Branchwidth
    • Forbidden
    • Minimal
    • Matroid
    • Minors
    • Graphic
    • Graphs
    • Tree
    • Treewidth
    • Width
    • Binary
    • Carving
  • undirected graph
    • Branchwidth
    • Forbidden
    • Minimal
    • Matroid
    • Minors
    • Graphic
    • Graphs
    • Tree
    • Treewidth
    • Width
    • Binary
    • Carving
  • unrooted binary tree
    • Unrooted
    • Tree
    • Leaves
    • Edges
    • Branch-decomposition
    • Carving
    • Graph
    • Two
    • Algorithms
    • Number
    • Width
    • Decomposition
  • planar graphs
    • Exactly
    • Time
    • Polynomial
    • May
    • Planar
    • Treewidth
    • Branch-decompositions
    • Algorithm
    • Branch
    • Constant
    • Forbidden
    • Particular
  • medial graph
    • Branchwidth
    • Forbidden
    • Minimal
    • Matroid
    • Minors
    • Graphic
    • Graphs
    • Tree
    • Treewidth
    • Width
    • Binary
    • Carving
  • minor-closed family of graphs
    • May
    • Planar
    • Time
    • Branch-decompositions
    • Constant
    • Forbidden
    • Matroids
    • Treewidth
    • Minors
    • Algorithm
    • Polynomial
    • Exactly
  • path graph
    • Branchwidth
    • Forbidden
    • Minimal
    • Matroid
    • Minors
    • Graphic
    • Graphs
    • Tree
    • Treewidth
    • Width
    • Binary
    • Carving
  • graph minors
    • Branchwidth
    • Minimal
    • Forbidden
    • Matroid
    • Minors
    • Graphic
    • Graphs
    • Matroids
    • Tree
    • Treewidth
    • Width
    • Binary

Connections between topic areas Semantic bridges

For Branch-decomposition, one of the stronger structural bridges in this analysis connects Branch-decomposition with Forbidden minors. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Branch-decomposition — Forbidden minors · splits 44 ⟂ 14
Branch-decomposition — Algorithms and complexity · splits 45 ⟂ 13
Branch-decomposition — Generalization to matroids · splits 46 ⟂ 12
Branch-decomposition — Overview · splits 48 ⟂ 10
Branch-decomposition — Relation to treewidth · splits 54 ⟂ 4

Map overview Semantic statistics

Branch-decomposition

Nodes58
Edges57
Triples32
Avg. degree1.97
Density0.034483
Components1

Source & methodology

TTTA analyzes the structure around Branch-decomposition to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Art, Forbidden minors & Algorithms and complexity, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Branch-decomposition · EN edition · Analysis: TopicsToTalkAbout

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