What question did this study set out to answer?

This note focuses on characterizing extremal trees that achieve specific bounds for total domination in graphs.

May 9, 2026Discrete Mathematics Algorithms and Applications0 citations

Characterizations of Total Domination in R ( T )

GVGayathri Kulandai VadivelSASTRA University KRKalaiyarasi RaviSASTRA University YVYanamandram B. VenkatakrishnanSASTRA University

Key Points

This note focuses on characterizing extremal trees that achieve specific bounds for total domination in graphs.
Defined total domination number in terms of vertex sets and neighbors.
Constructed graphs by adding vertices based on edges of the original graph.
Analyzed properties of extremal trees in relation to total domination.
Provided a constructive characterization of extremal trees achieving the bounds proved by Bermudo.
Confirmed that for trees with a certain number of leaves, specific total domination numbers apply.

Abstract

A set Formula: see text of vertices is a total dominating set of a graph Formula: see text if every vertex of Formula: see text has at least one neighbor in Formula: see text. The total domination number Formula: see text is the minimum cardinality among all total dominating sets of Formula: see text The graph Formula: see text is obtained by adding a new vertex Formula: see text for each edge Formula: see text and joining Formula: see text to the end vertices Formula: see text and Formula: see text of Formula: see text. S. Bermudo, Total domination on tree operators, Mediterr. J. Math. 20 (2023) 42. proved that for every tree Formula: see text of order Formula: see text with Formula: see text leaves, Formula: see text. In this note, we provide a constructive characterization of extremal trees attaining these two bounds.

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