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    We introduce new quality metrics for symmetric graph drawing, measuring how faithfully drawings display graph symmetries. These metrics, based on group theory, evaluate both single and group automorphisms for better graph visualization.

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    Area of Science:

    • Graph theory
    • Computational geometry
    • Group theory

    Background:

    • Symmetric graph drawing aims to represent graph automorphisms as geometric symmetries.
    • Existing methods lack precise quantitative measures for evaluating drawing faithfulness to graph symmetries.

    Purpose of the Study:

    • To introduce novel quality metrics for symmetric graph drawing.
    • To quantify the faithfulness of a graph drawing to its underlying geometric automorphisms.
    • To evaluate and compare existing graph drawing algorithms based on symmetry preservation.

    Main Methods:

    • Development of automorphism faithfulness metrics based on group theory.
    • Algorithms for computing metrics in O(n log n) time.
    • Efficient algorithms for detecting exact symmetries in graph drawings.
    • Validation through deformation experiments and evaluation of existing algorithms.

    Main Results:

    • Introduction of two types of automorphism faithfulness metrics: single automorphism (axial/rotational) and group (cyclic/dihedral).
    • Efficient algorithms achieve O(n log n) time complexity for metric computation.
    • Experimental validation confirms the utility of the metrics in assessing drawing fidelity.
    • Comparison of existing graph drawing algorithms reveals varying degrees of symmetry preservation.

    Conclusions:

    • The proposed metrics provide a quantitative framework for evaluating symmetric graph drawings.
    • These metrics enable objective comparison of graph drawing algorithms regarding symmetry representation.
    • The findings contribute to the development of more aesthetically pleasing and informative graph visualizations.