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Total domination subdivision numbers of trees. (English) Zbl 1054.05076

The total domination subdivision number \(\text{ sd}_{\gamma_t}(G)\) of a graph \(G\) is the minimum number of edges whose subdivision increases the total domination number \({\gamma_t}(G)\) of \(G\). T. W. Haynes et al. [J. Comb. Math. Comb. Comput. 44, 115–128 (2003; Zbl 1020.05048)] have shown that \(1\leq \text{ sd}_{\gamma_t}(T)\leq 3\) for any tree \(T\). In the present paper the authors provide a constructive characterization of the family \({\mathcal F}\) of trees \(T\) with \(\text{ sd}_{\gamma_t}(T)=3\). The family \({\mathcal F}\) consists of labeled trees, contains a path of order \(6\) whose vertices have labels \(c,b,a,a,b,c\) and is closed under the two operations \({\mathcal T}_1\) and \({\mathcal T}_2\) where \({\mathcal T}_1\) consists of adding a path of order \(3\) labeled \(a,b,c\) to a tree \(T\in {\mathcal F}\) and joining the vertex labeled \(a\) to a vertex labeled \(a\) in \(T\) and \({\mathcal T}_2\) consists of adding a path of order \(4\) labeled \(a,a,b,c\) to a tree \(T\in {\mathcal F}\) and joining the endvertex labeled \(a\) to a vertex labeled \(b\) or \(c\) in \(T\).

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C05 Trees

Citations:

Zbl 1020.05048
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References:

[1] Cockayne, E.J.; Dawes, R.M.; Hedetniemi, S.T., Total domination in graphs, Networks, 10, 211-219, (1980) · Zbl 0447.05039
[2] Haynes, T.W.; Hedetniemi, S.M.; Hedetniemi, S.T., Domination and independence subdivision numbers of graphs, Discuss. math. graph theory, 20, 271-280, (2000) · Zbl 0984.05066
[3] Haynes, T.W.; Hedetniemi, S.T.; Slater, P.J., Fundamentals of domination in graphs, (1998), Marcel Dekker New York · Zbl 0890.05002
[4] T.W. Haynes, S.T. Hedetniemi, P.J. Slater (Eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998. · Zbl 0883.00011
[5] Haynes, T.W.; Hedetniemi, S.T.; van der Merwe, L.C., Total domination subdivision numbers, J. combin. math. combin. comput., 44, 115-128, (2003) · Zbl 1020.05048
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