On trees attaining an upper bound on the total domination number

Document Type : Research Paper


Department of Pure and Applied Mathematics, University of Johannesburg, South Africa \newline Research fellow of the Claude Leon Foundation. Faculty of Electronics, Telecommunications and Informatics, Gdansk University of Technology, Poland.


‎A total dominating set of a graph $G$ is a set $D$ of vertices of $G$ such that every vertex of $G$ has a neighbor in $D$‎. ‎The total domination number of a graph $G$‎, ‎denoted by $\gamma_t(G)$‎, ‎is~the minimum cardinality of a total dominating set of $G$‎. ‎Chellali and Haynes [Total and paired-domination numbers of a tree, AKCE International ournal of Graphs and Combinatorics 1 (2004)‎, ‎69--75] established the following upper bound on the total domination number of a tree in terms of the order and the number of support vertices‎, ‎$\gamma_t(T) \le (n+s)/2$‎. ‎We characterize all trees attaining this upper bound‎.‎


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