Bulletin of the
Korean Mathematical Society BKMS

Let $G$ be a graph with no isolated vertex. \emph{A total dominating set}, abbreviated TDS of $G$ is a subset $S$ of vertices of $G$ such that every vertex of $G$ is adjacent to a vertex in $S$. \emph{The total domination number} of $G$ is the minimum cardinality of a TDS of $G$. In this paper, we study the total domination number of central graphs. Indeed, we obtain some tight bounds for the total domination number of a central graph $C(G)$ in terms of some invariants of the graph $G$. Also we characterize the total domination number of the central graph of some families of graphs such as path graphs, cycle graphs, wheel graphs, complete graphs and complete multipartite graphs, explicitly. Moreover, some Nordhaus-Gaddum-like relations are presented for the total domination number of central graphs.

Keywords: total domination number, central graph, Nordhaus-Gaddum-like relation

MSC numbers: Primary 05C76; Secondary 97K30