Total Domination Index in Graphs

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 classes.