### Bounds on weak and strong total domination in graphs

#### Abstract

A set $D$ of vertices in a graph $G=(V,E)$ is a total dominating

set if every vertex of $G$ is adjacent to some vertex in $D$. A

total dominating set $D$ of $G$ is said to be weak if every

vertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that

$d_{G}(v)\geq d_{G}(u)$. The weak total domination number

$\gamma_{wt}(G)$ of $G$ is the minimum cardinality of a weak

total dominating set of $G$. A total dominating set $D$ of $G$ is

said to be strong if every vertex $v\in V-D$ is adjacent to a

vertex $u\in D$ such that $d_{G}(v)\leq d_{G}(u)$. The strong

total domination number $\gamma_{st}(G)$ of $G$ is the minimum

cardinality of a strong total dominating set of $G$. We present

some bounds on weak and strong total domination number of a graph.

set if every vertex of $G$ is adjacent to some vertex in $D$. A

total dominating set $D$ of $G$ is said to be weak if every

vertex $v\in V-D$ is adjacent to a vertex $u\in D$ such that

$d_{G}(v)\geq d_{G}(u)$. The weak total domination number

$\gamma_{wt}(G)$ of $G$ is the minimum cardinality of a weak

total dominating set of $G$. A total dominating set $D$ of $G$ is

said to be strong if every vertex $v\in V-D$ is adjacent to a

vertex $u\in D$ such that $d_{G}(v)\leq d_{G}(u)$. The strong

total domination number $\gamma_{st}(G)$ of $G$ is the minimum

cardinality of a strong total dominating set of $G$. We present

some bounds on weak and strong total domination number of a graph.

#### Keywords

weak total domination, strong total domination, Nordhaus-Gaddum

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2016.4.1.10

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