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Related Experiment Videos

Enumeration of structure-sensitive graphical subsets: Calculations.

R E Marrifield1, H E Simmons

  • 1Central Research and Development Department, E. I. du Pont de Nemours and Company, Experimental Station, Wilmington, Delaware 19898.

Proceedings of the National Academy of Sciences of the United States of America
|March 1, 1981
PubMed
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This study numerically calculates various graph set counts for graphs with six or fewer vertices. Most set counts, excluding kernels, are highly sensitive to graph structure.

Area of Science:

  • Graph theory
  • Discrete mathematics
  • Combinatorics

Background:

  • Graph theory is a fundamental area of discrete mathematics.
  • Understanding the properties of sets within graphs is crucial for various applications.
  • Previous research has explored specific graph properties, but comprehensive calculations for smaller graphs are needed.

Purpose of the Study:

  • To numerically calculate the number of independent sets, connected sets, point and line covers, externally stable sets, kernels, and irredundant sets.
  • To analyze all connected graphs with six or fewer vertices.
  • To determine the structure-sensitivity of these set counts.

Main Methods:

  • Utilized numerical calculations.
  • Examined all connected graphs up to six vertices.

Related Experiment Videos

  • Quantified specific types of sets within these graphs.

    • Main Results:

    • Presented numerical calculations for independent sets, connected sets, point and line covers, externally stable sets, kernels, and irredundant sets.
    • Demonstrated that the number of these sets, except for kernels, are highly structure-sensitive quantities.
    • Provided a comprehensive enumeration for graphs with n <= 6.

    Conclusions:

    • The number of independent sets, connected sets, point and line covers, externally stable sets, and irredundant sets are significantly influenced by graph structure.
    • Kernel counts exhibit different behavior compared to other analyzed set types.
    • This enumeration provides a foundational dataset for further graph theory research.