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• # Artykuł - szczegóły

## Discussiones Mathematicae Graph Theory

2016 | 36 | 1 | 153-171

## Bounds On The Disjunctive Total Domination Number Of A Tree

EN

### Abstrakty

EN
Let G be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, γt(G). A set S of vertices in G is a disjunctive total dominating set of G if every vertex is adjacent to a vertex of S or has at least two vertices in S at distance 2 from it. The disjunctive total domination number, [...] γtd(G) $\gamma _t^d (G)$ , is the minimum cardinality of such a set. We observe that [...] γtd(G)≤γt(G) $\gamma _t^d (G) \le \gamma _t (G)$ . A leaf of G is a vertex of degree 1, while a support vertex of G is a vertex adjacent to a leaf. We show that if T is a tree of order n with ℓ leaves and s support vertices, then [...] 2(n−ℓ+3)/5≤γtd(T)≤(n+s−1)/2 $2(n - \ell + 3)/5 \le \gamma _t^d (T) \le (n + s - 1)/2$ and we characterize the families of trees which attain these bounds. For every tree T, we show have [...] γt(T)/γtd(T)<2 $\gamma _t (T)/\gamma _t^d (T) < 2$ and this bound is asymptotically tight.

EN

153-171

wydano
2016-02-01
poprawiono
2015-05-28
zaakceptowano
2015-05-28
otrzymano
2015-11-14
online
2016-01-19

### Twórcy

autor
• Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa
autor
• Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, 2006, South Africa; and Department of Mathematics, Rhodes University, Grahamstown, 6140 South Africa

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