Upper k-tuple total domination in graphs

TL;DR

This paper introduces the concept of upper k-tuple total domination number in graphs, exploring its properties, bounds, and behavior on various graph classes and graph products.

Contribution

It defines the upper k-tuple total domination number and investigates its properties, bounds, and specific values for different classes of graphs and graph products.

Findings

Defined the upper k-tuple total domination number.

Established bounds and characterizations for this parameter.

Analyzed the parameter for Cartesian and cross product graphs.

Abstract

Let $G=(V,E)$ be a simple graph. For any integer $k\geq 1$, a subset of $V$ is called a $k$-tuple total dominating set of $G$ if every vertex in $V$ has at least $k$ neighbors in the set. The minimum cardinality of a minimal $k$-tuple total dominating set of $G$ is called the $k$-tuple total domination number of $G$. In this paper, we introduce the concept of upper $k$-tuple total domination number of $G$ as the maximum cardinality of a minimal $k$-tuple total dominating set of $G$, and study the problem of finding a minimal $k$-tuple total dominating set of maximum cardinality on several classes of graphs, as well as finding general bounds and characterizations. Also, we find some results on the upper $k$-tuple total domination number of the Cartesian and cross product graphs.