## 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

### Keywords:

total domination; total domination subdivision number

Zbl 1020.05048
Full Text:

### 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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