# Trees with equal total domination and total restrained domination numbers

Xue-Gang Chen; Wai Chee Shiu; Hong-Yu Chen

Discussiones Mathematicae Graph Theory (2008)

- Volume: 28, Issue: 1, page 59-66
- ISSN: 2083-5892

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topXue-Gang Chen, Wai Chee Shiu, and Hong-Yu Chen. "Trees with equal total domination and total restrained domination numbers." Discussiones Mathematicae Graph Theory 28.1 (2008): 59-66. <http://eudml.org/doc/270189>.

@article{Xue2008,

abstract = {For a graph G = (V,E), a set S ⊆ V(G) is a total dominating set if it is dominating and both ⟨S⟩ has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V(G) is a total restrained dominating set if it is total dominating and ⟨V(G)-S⟩ has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same.},

author = {Xue-Gang Chen, Wai Chee Shiu, Hong-Yu Chen},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {total domination number; total restrained domination number; tree},

language = {eng},

number = {1},

pages = {59-66},

title = {Trees with equal total domination and total restrained domination numbers},

url = {http://eudml.org/doc/270189},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Xue-Gang Chen

AU - Wai Chee Shiu

AU - Hong-Yu Chen

TI - Trees with equal total domination and total restrained domination numbers

JO - Discussiones Mathematicae Graph Theory

PY - 2008

VL - 28

IS - 1

SP - 59

EP - 66

AB - For a graph G = (V,E), a set S ⊆ V(G) is a total dominating set if it is dominating and both ⟨S⟩ has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V(G) is a total restrained dominating set if it is total dominating and ⟨V(G)-S⟩ has no isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same.

LA - eng

KW - total domination number; total restrained domination number; tree

UR - http://eudml.org/doc/270189

ER -

## References

top- [1] S. Arumugam and J. Paulraj Joseph, On graphs with equal domination and connected domination numbers, Discrete Math. 206 (1999) 45-49, doi: 10.1016/S0012-365X(98)00390-2. Zbl0933.05114
- [2] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi, R.C. Laskar and L.R. Marcus, Restrained domination in graphs, Discrete Math. 203 (1999) 61-69, doi: 10.1016/S0012-365X(99)00016-3. Zbl1114.05303
- [3] F. Harary and M. Livingston, Characterization of tree with equal domination and independent domination numbers, Congr. Numer. 55 (1986) 121-150. Zbl0647.05020
- [4] D. Ma, X. Chen and L. Sun, On total restrained domination in graphs, Czechoslovak Math. J. 55 (2005) 165-173, doi: 10.1007/s10587-005-0012-2. Zbl1081.05086
- [5] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi and L.R. Markus, Restrained domination in trees, Discrete Math. 211 (2000) 1-9, doi: 10.1016/S0012-365X(99)00036-9. Zbl0947.05057
- [6] E.J. Cockayne, C.M. Mynhardt and B. Yu, Total dominating functions in trees: minimality and convexity, J. Graph Theory 19 (1995) 83-92, doi: 10.1002/jgt.3190190109. Zbl0819.05035

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