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# Pure and Applied Mathematics Quarterly

## Volume 13 (2017)

### Number 4

### Upper $k$-tuple total domination in graphs

Pages: 563 – 579

DOI: https://dx.doi.org/10.4310/PAMQ.2017.v13.n4.a1

#### Author

#### 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.

#### Keywords

$k$-tuple total domination number, upper $k$-tuple total domination number, Cartesian and cross product graphs, hypergraph, (upper) $k$-transversal number

#### 2010 Mathematics Subject Classification

05C69

Received 1 February 2018

Published 21 December 2018