# Total outer-connected domination in trees

• Volume: 30, Issue: 3, page 377-383
• ISSN: 2083-5892

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## Abstract

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Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by ${\gamma }_{tc}\left(G\right)$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then ${\gamma }_{tc}\left(T\right)\ge ⎡2n/3⎤$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.

## How to cite

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Joanna Cyman. "Total outer-connected domination in trees." Discussiones Mathematicae Graph Theory 30.3 (2010): 377-383. <http://eudml.org/doc/270887>.

@article{JoannaCyman2010,
abstract = {Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by $γ_\{tc\}(G)$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then $γ_\{tc\}(T) ≥ ⎡2n/3⎤$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.},
author = {Joanna Cyman},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {total outer-connected domination number; domination number},
language = {eng},
number = {3},
pages = {377-383},
title = {Total outer-connected domination in trees},
url = {http://eudml.org/doc/270887},
volume = {30},
year = {2010},
}

TY - JOUR
AU - Joanna Cyman
TI - Total outer-connected domination in trees
JO - Discussiones Mathematicae Graph Theory
PY - 2010
VL - 30
IS - 3
SP - 377
EP - 383
AB - Let G = (V,E) be a graph. Set D ⊆ V(G) is a total outer-connected dominating set of G if D is a total dominating set in G and G[V(G)-D] is connected. The total outer-connected domination number of G, denoted by $γ_{tc}(G)$, is the smallest cardinality of a total outer-connected dominating set of G. We show that if T is a tree of order n, then $γ_{tc}(T) ≥ ⎡2n/3⎤$. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.
LA - eng
KW - total outer-connected domination number; domination number
UR - http://eudml.org/doc/270887
ER -

## References

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1. [1] G. Chartrand and L. Leśniak, Graphs & Digraphs (Wadsworth and Brooks/Cole, Monterey, CA, third edition, 1996).
2. [2] E.J. Cockayne, R.M. Dawes and S.T. Hedetniemi, Total domination in graphs, Networks 10 (1980) 211-219, doi: 10.1002/net.3230100304. Zbl0447.05039
3. [3] G.S. Domke, J.H. Hattingh, M.A. Henning and L.R. Markus, Restrained domination in trees, Discrete Math. 211 (2000) 1-9, doi: 10.1016/S0012-365X(99)00036-9. Zbl0947.05057
4. [4] J.H. Hattingh, E. Jonck, E.J. Joubert and A.R. Plummer, Total Restrained Domination in Trees, Discrete Math. 307 (2007) 1643-1650, doi: 10.1016/j.disc.2006.09.014. Zbl1132.05044
5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
6. [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998). Zbl0883.00011

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