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Penny graph: Measurement, Properties & Overview

In geometric graph theory, a penny graph is a contact graph of unit circles. It is formed from a collection of unit circles that do not cross each other, by creating a vertex for each circle and an edge for every pair of tangent circles. The circles can be represented physically by pennies, arranged without overlapping on a flat surface, with a vertex…

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Penny graph topic overview

The analysis highlights Measurement, Properties and Overview as prominent areas in the source structure around Penny graph.

Related topics
36
Source areas
4
Connected nodes
40
Extracted relationships
42
Concept neighborhoods
23
Bridge connections
40

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.

Overview · 16 topics
Properties · 13 topics
Computational complexity · 5 topics
Related graph families · 2 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

Properties

Computational complexity

Related graph families

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 Penny graph connects Entity context

The extracted context around Penny graph shows recurring relationship patterns in the source. For example, Penny graph → As, Baker's, By, Finding, However, In, It, NP-hard, Pach, Paul Erdős, Swanepoel, That, Tóth Another extracted example is Penny graph → Analogously, Based, Every, For, Grötzsch's, However, Therefore. Use these groups to spot repeated connection types before inspecting the individual relationships.

Penny graph

Top relations

related to Independent sets · 13
Penny graph → As, Baker's, By, Finding, However, In, It, NP-hard, Pach, Paul Erdős, Swanepoel, That, Tóth
related to Coloring · 7
Penny graph → Analogously, Based, Every, For, Grötzsch's, However, Therefore
related to Computational complexity · 7
Penny graph → An, Constructing, Delaunay, However, It, NP-hard, Similarly
related to Number of edges · 7
Penny graph → By, Counting, Every, However, Proving, Some, Swanepoel
is a · 3
Penny graph → contact graph of unit circles, subset of the pennies, unit disk graph and a matchstick graph
related to Related graph families · 3
Penny graph → Because, Penny, The

Important terminology

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

Important terminology

penny graph graphs circles vertex every unit edges pennies np-hard represented two planar number independent one vertices neighbors however circle

Penny graph relationships Subject–Predicate–Object triples

TTTA extracted 42 structured relationships around Penny graph. Examples in this analysis include Penny graph → is a → contact graph of unit circles and Penny graph → is a → unit disk graph and a matchstick graph. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Penny graphis acontact graph of unit circles0.90text
Penny graphis aunit disk graph and a matchstick graph0.90text
Penny graphis asubset of the pennies0.90text
testing adjacencyinstance ofgiven an input representing its circles in a form allowing basic computational tasks0.80text
finding intersections of the circles with axis-parallel linesinstance ofgiven an input representing its circles in a form allowing basic computational tasks0.80text
Penny graphrelated to ColoringEvery0.60section
Penny graphrelated to ColoringFor0.60section
Penny graphrelated to ColoringTherefore0.60section
Penny graphrelated to ColoringBased0.60section
Penny graphrelated to ColoringHowever0.60section
Penny graphrelated to ColoringAnalogously0.60section
Penny graphrelated to ColoringGrötzsch's0.60section

Related concept clusters Concept neighborhoods

The concept neighborhoods around Penny graph bring nearby vocabulary together. In this analysis, examples include Penny, Graphs and Edges. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Penny graph
    • Penny
    • Graphs
    • Edges
    • Independent
    • Number
    • Every
    • Pennies
    • Nearest
    • Neighbor
    • Unit
    • Vertices
    • Vertex
  • penny graph
    • Penny
    • Graphs
    • Every
    • Edges
    • Independent
    • Neighbors
    • Np-hard
    • Number
    • Vertices
    • Pennies
    • Nearest
    • Neighbor
  • geometric graph theory
    • Penny
    • Every
    • Edges
    • Independent
    • Neighbors
    • Np-hard
    • Number
    • Vertices
    • Nearest
    • Neighbor
    • Set
    • Testing
  • contact graph
    • Penny
    • Every
    • Edges
    • Independent
    • Neighbors
    • Np-hard
    • Number
    • Vertices
    • Nearest
    • Neighbor
    • Set
    • Testing
  • unit circles
    • Unit
    • Edge
    • Vertex
    • Penny
    • Also
    • Coin
    • Formed
    • Graphs
    • Plane
    • Represented
    • Edges
    • Every
  • tangent circles
    • Unit
    • Edge
    • Vertex
    • Penny
    • Coin
    • Formed
    • Graphs
    • Also
    • Plane
    • Represented
    • Edges
    • Every
  • coin graphs
    • Penny
    • Planar
    • Arbitrary
    • Coin
    • Formed
    • Graphs
    • Also
    • Finding
    • However
    • One
    • Edges
    • Represented
  • unit disk graph
    • Penny
    • Every
    • Also
    • Edges
    • Independent
    • Neighbors
    • Np-hard
    • Number
    • Vertices
    • Nearest
    • Neighbor
    • Set

Connections between topic areas Semantic bridges

For Penny graph, one of the stronger structural bridges in this analysis connects Penny graph with Overview. 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
Penny graph — Overview · splits 24 ⟂ 17
Penny graph — Properties · splits 27 ⟂ 14
Penny graph — Computational complexity · splits 35 ⟂ 6
Penny graph — Related graph families · splits 38 ⟂ 3

Map overview Semantic statistics

Penny graph

Nodes41
Edges40
Triples42
Avg. degree1.95
Density0.04878
Components1

Source & methodology

TTTA analyzes the structure around Penny graph to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Measurement, Properties & Overview, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Penny graph · EN edition · Analysis: TopicsToTalkAbout

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