# On locating and differentiating-total domination in trees

• Volume: 28, Issue: 3, page 383-392
• ISSN: 2083-5892

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## Abstract

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A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $\gamma {ₜ}^{L}\left(G\right)$ and $\gamma {ₜ}^{D}\left(G\right)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with l leaves and s support vertices, $\gamma {ₜ}^{L}\left(T\right)\ge max2\left(n+l-s+1\right)/5,\left(n+2-s\right)/2$, and for a tree of order n ≥ 3, $\gamma {ₜ}^{D}\left(T\right)\ge 3\left(n+l-s+1\right)/7$, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying $\gamma {ₜ}^{L}\left(T\right)=2\left(n+l-s+1\right)/5$ or $\gamma {ₜ}^{D}\left(T\right)=3\left(n+l-s+1\right)/7$.

## How to cite

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Mustapha Chellali. "On locating and differentiating-total domination in trees." Discussiones Mathematicae Graph Theory 28.3 (2008): 383-392. <http://eudml.org/doc/270636>.

@article{MustaphaChellali2008,
abstract = {A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $γₜ^L(G)$ and $γₜ^D(G)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with l leaves and s support vertices, $γₜ^L(T) ≥ max\{2(n+l-s+1)/5,(n+2-s)/2\}$, and for a tree of order n ≥ 3, $γₜ^D(T) ≥ 3(n+l-s+1)/7$, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying $γₜ^L(T) = 2(n+l- s+1)/5$ or $γₜ^D(T) = 3(n+l-s+1)/7$.},
author = {Mustapha Chellali},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {locating-total domination; differentiating-total domination; trees},
language = {eng},
number = {3},
pages = {383-392},
title = {On locating and differentiating-total domination in trees},
url = {http://eudml.org/doc/270636},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Mustapha Chellali
TI - On locating and differentiating-total domination in trees
JO - Discussiones Mathematicae Graph Theory
PY - 2008
VL - 28
IS - 3
SP - 383
EP - 392
AB - A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $γₜ^L(G)$ and $γₜ^D(G)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total dominating set of G, respectively. We show that for a nontrivial tree T of order n, with l leaves and s support vertices, $γₜ^L(T) ≥ max{2(n+l-s+1)/5,(n+2-s)/2}$, and for a tree of order n ≥ 3, $γₜ^D(T) ≥ 3(n+l-s+1)/7$, improving the lower bounds of Haynes, Henning and Howard. Moreover we characterize the trees satisfying $γₜ^L(T) = 2(n+l- s+1)/5$ or $γₜ^D(T) = 3(n+l-s+1)/7$.
LA - eng
KW - locating-total domination; differentiating-total domination; trees
UR - http://eudml.org/doc/270636
ER -

## References

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1. [1] M. Blidia, M. Chellali, F. Maffray, J. Moncel and A. Semri, Locating-domination and identifying codes in trees, Australasian J. Combin. 39 (2007) 219-232. Zbl1136.05049
2. [2] M. Chellali and T.W. Haynes, A note on the total domination number of a tree, J. Combin. Math. Combin. Comput. 58 (2006) 189-193. Zbl1114.05071
3. [3] J. Gimbel, B. van Gorden, M. Nicolescu, C. Umstead and N. Vaiana, Location with dominating sets, Congr. Numer. 151 (2001) 129-144. Zbl0996.05095
4. [4] T.W. Haynes, M.A. Henning and J. Howard, Locating and total dominating sets in trees, Discrete Appl. Math. 154 (2006) 1293-1300, doi: 10.1016/j.dam.2006.01.002. Zbl1091.05051

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