Total Domination Index in Graphs
Kavya R. Nair, M. S. Sunitha

TL;DR
This paper introduces the total domination degree and total domination index in graphs, providing formulas, bounds, and properties for various graph classes and operations, advancing understanding of domination concepts.
Contribution
It defines and systematically analyzes the total domination degree and index, including their behavior under graph operations and in different graph classes, which is a novel contribution.
Findings
Formulas for TDD in various graph classes
Bounds for TDI across graph classes
Properties of graphs under operations like union and join
Abstract
This paper introduces the concept of compliant vertices and compliant graphs, with a focus on the total domination degree (TDD) of a vertex in compliant graphs. The TDD is systematically calculated for various graph classes, including path graphs, cycles, book graphs, windmill graphs, wheel graphs, complete graphs, and complete bipartite graphs. The study explores inequalities involving TDD and defines total domination regular graphs. Furthermore, the TDD is analyzed in several graph operations such as union, join, composition, and corona, with a discussion on the property of the resulting graphs. The paper also examines the subdivision of complete graphs and degree splitting of path graphs. In the subsequent section, the total domination index (TDI) is introduced, and its values are calculated for different graph classes. The study concludes with bounds for the TDI across these graph…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Graph theory and applications
