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Cycle basis: Applications, Special cycle bases & Minimum weight

In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles.

Language: English [EN]
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Cycle basis topic overview

The analysis highlights Applications, Special cycle bases and Minimum weight as prominent areas in the source structure around Cycle basis.

Related topics
66
Source areas
5
Connected nodes
71
Extracted relationships
60
Concept neighborhoods
31
Bridge connections
71

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.

Special cycle bases · 20 topics
Minimum weight · 14 topics
Overview · 13 topics
Applications · 11 topics
Definitions · 8 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

Special cycle bases

Minimum weight

Applications

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 Cycle basis connects Entity context

The extracted context around Cycle basis shows recurring relationship patterns in the source. For example, Cycle basis → As, By Euler's, Eulerian, For, If, In, It, Mac Lane's, One, The Another extracted example is Cycle basis → As Horton, Gaussian, His, Horton, However, In, Subsequent, Testing, There, Using Dijkstra's. Use these groups to spot repeated connection types before inspecting the individual relationships.

Cycle basis

Top relations

related to Face cycles · 10
Cycle basis → As, By Euler's, Eulerian, For, If, In, It, Mac Lane's, One, The
related to Polynomial time algorithms · 10
Cycle basis → As Horton, Gaussian, His, Horton, However, In, Subsequent, Testing, There, Using Dijkstra's
related to In planar graphs · 6
Cycle basis → Based, Gomory, However, Hu, The, This
related to Minimum weight · 6
Cycle basis → By, Eulerian, For, If, The, Veblen's
related to Definitions · 5
Cycle basis → Eulerian, Every, It, The, This
related to Fundamental cycles · 5
Cycle basis → Each, If, It, There, Therefore
related to NP-hardness · 5
Cycle basis → Finding, Hamiltonian, If, MAXSNP-hard, NP-hard
related to Integral bases · 4
Cycle basis → An, Less, The, This
is a · 3
Cycle basis → basis of this vector space in which each basis vector represents a simple cycle, Horton cycle, set of simple cycles that generates this group
has application · 2
Cycle basis → Cycle, In

Important terminology

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

Important terminology

cycle basis graph cycles weight fundamental minimum set displaystyle edge edges planar every given bases space may tree one eulerian

Cycle basis relationships Subject–Predicate–Object triples

TTTA extracted 60 structured relationships around Cycle basis. Examples in this analysis include Cycle basis → is a → basis of this vector space in which each basis vector represents a simple cycle and Cycle basis → is a → set of simple cycles that generates this group. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Cycle basisis abasis of this vector space in which each basis vector represents a simple cycle0.90text
Cycle basisis aset of simple cycles that generates this group0.90text
Cycle basisis aHorton cycle0.90text
Cycle basishas applicationCycle0.60section
Cycle basishas applicationIn0.60section
Cycle basisrelated to DefinitionsEulerian0.60section
Cycle basisrelated to DefinitionsEvery0.60section
Cycle basisrelated to DefinitionsThe0.60section
Cycle basisrelated to DefinitionsIt0.60section
Cycle basisrelated to DefinitionsThis0.60section
Cycle basisrelated to Face cyclesIf0.60section
Cycle basisrelated to Face cyclesOne0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Cycle basis bring nearby vocabulary together. In this analysis, examples include Basis, Cycle and Graph. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Cycle basis
    • Basis
    • Cycle
    • Graph
    • Cycles
    • Fundamental
    • Weight
    • Minimum
    • Bases
    • Edge
    • Planar
    • Every
    • Used
  • cycle basis
    • Basis
    • Cycle
    • Graph
    • Weight
    • Cycles
    • Minimum
    • Fundamental
    • Bases
    • Every
    • Set
    • Edge
    • Planar
  • graph theory
    • Edges
    • Planar
    • Given
    • Weight
    • Cycles
    • Minimum
    • May
    • Displaystyle
    • Set
    • Edge
    • Fundamental
    • Formed
  • undirected graph
    • Edges
    • Planar
    • Given
    • Weight
    • Cycles
    • Minimum
    • May
    • Displaystyle
    • Set
    • Edge
    • Fundamental
    • Formed
  • simple cycles
    • Set
    • Space
    • Graph
    • Fundamental
    • One
    • Minimum
    • Eulerian
    • Subgraph
    • Weight
    • Planar
    • Every
    • Given
  • basis
    • Cycle
    • Weight
    • Cycles
    • Minimum
    • Graph
    • Fundamental
    • Every
    • Set
    • Planar
    • Space
    • Time
    • Weakly
  • cycle space
    • Basis
    • Graph
    • Cycles
    • Fundamental
    • Weight
    • Minimum
    • Bases
    • Edge
    • Planar
    • Every
    • Used
    • Weakly
  • planar graph
    • Bounded
    • Edges
    • Embedding
    • Planar
    • Given
    • Weight
    • Cycles
    • Faces
    • Minimum
    • Set
    • May
    • Displaystyle

Connections between topic areas Semantic bridges

For Cycle basis, one of the stronger structural bridges in this analysis connects Cycle basis with Special cycle bases. 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
Cycle basisSpecial cycle bases · splits 51 ⟂ 21
Cycle basisMinimum weight · splits 57 ⟂ 15
Cycle basisOverview · splits 58 ⟂ 14
Cycle basisApplications · splits 60 ⟂ 12
Cycle basisDefinitions · splits 63 ⟂ 9

Map overview Semantic statistics

Cycle basis

Nodes72
Edges71
Triples60
Avg. degree1.97
Density0.027778
Components1

Source & methodology

TTTA analyzes the structure around Cycle basis to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Applications, Special cycle bases & Minimum weight, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Cycle basis · EN edition · Analysis: TopicsToTalkAbout

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