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Title:
On the total domination number of total graphs
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Discussiones Mathematicae Graph Theory 44(3) (2024) 933-951
Received: 2022-07-12 , Revised: 2022-11-08 , Accepted: 2022-11-12 , Available online: 2022-12-05 , https://doi.org/10.7151/dmgt.2478
Abstract:
Let $G$ be a graph with no isolated vertex. A set $D\subseteq V(G)$ is a total
dominating set of $G$ if every vertex of $G$ is adjacent to at least one vertex
in $D$. The total domination number of $G$, denoted by $\gamma_{t}(G)$, is the
minimum cardinality among all total dominating sets of $G$. In this paper we
study the total domination number of total graphs $\texttt{T}(G)$ of simple
graphs $G$. In particular, we give some relationships that exist between
$\gamma_{t}(\texttt{T}(G))$ and other domination parameters of $G$ and of some
well-known graph operators on $G$. Finally, we provide closed formulas on
$\gamma_t(\texttt{T}(G))$ for some well-known families of graphs $G$.
Keywords:
total domination, graph operators, total graphs
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