Total domination number of central graphs
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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.
Abstract
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.
참고문헌 (6)
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- M. A. Henning, "Total Domination in Graphs, Springer Monographs in Math-ematics" (2013), Springer
- E. A. Nordhaus, "On Complementary Graphs", The American Mathematical Monthly, 63(3), pp.175~177 (1956), Informa UK Limited[DOI]
- J. V. Vernold, "Harmonious coloring of total graphs, n-leaf, central graphs and circumdetic graphs", Ph.D, pp.2007
- D. B. West, "Introduction to Graph Theory" (1996), Prentice Hall, Inc.