# Trees with equal restrained domination and total restrained domination numbers

• Volume: 27, Issue: 1, page 83-91
• ISSN: 2083-5892

top

## Abstract

top
For a graph G = (V,E), a set D ⊆ V(G) is a total restrained dominating set if it is a dominating set and both ⟨D⟩ and ⟨V(G)-D⟩ do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V(G) is a restrained dominating set if it is a dominating set and ⟨V(G)-D⟩ does not contain an isolated vertex. The cardinality of a minimum restrained dominating set in G is the restrained domination number. We characterize all trees for which total restrained and restrained domination numbers are equal.

## How to cite

top

Joanna Raczek. "Trees with equal restrained domination and total restrained domination numbers." Discussiones Mathematicae Graph Theory 27.1 (2007): 83-91. <http://eudml.org/doc/270392>.

@article{JoannaRaczek2007,
abstract = {For a graph G = (V,E), a set D ⊆ V(G) is a total restrained dominating set if it is a dominating set and both ⟨D⟩ and ⟨V(G)-D⟩ do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V(G) is a restrained dominating set if it is a dominating set and ⟨V(G)-D⟩ does not contain an isolated vertex. The cardinality of a minimum restrained dominating set in G is the restrained domination number. We characterize all trees for which total restrained and restrained domination numbers are equal.},
author = {Joanna Raczek},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {total restrained domination number; restrained domination number; trees; total restrained set; restrained dominating set},
language = {eng},
number = {1},
pages = {83-91},
title = {Trees with equal restrained domination and total restrained domination numbers},
url = {http://eudml.org/doc/270392},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Joanna Raczek
TI - Trees with equal restrained domination and total restrained domination numbers
JO - Discussiones Mathematicae Graph Theory
PY - 2007
VL - 27
IS - 1
SP - 83
EP - 91
AB - For a graph G = (V,E), a set D ⊆ V(G) is a total restrained dominating set if it is a dominating set and both ⟨D⟩ and ⟨V(G)-D⟩ do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V(G) is a restrained dominating set if it is a dominating set and ⟨V(G)-D⟩ does not contain an isolated vertex. The cardinality of a minimum restrained dominating set in G is the restrained domination number. We characterize all trees for which total restrained and restrained domination numbers are equal.
LA - eng
KW - total restrained domination number; restrained domination number; trees; total restrained set; restrained dominating set
UR - http://eudml.org/doc/270392
ER -

## References

top
1. [1] 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
2. [2] G.S. Domke, J.H. Hattingh, S.T. Hedetniemi and L.R. Marcus, Restrained domination in trees, Discrete Math. 211 (2000) 1-9, doi: 10.1016/S0012-365X(99)00036-9.
3. [3] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
4. [4] M.A. Henning, Trees with equal average domination and independent domination numbers, Ars Combin. 71 (2004) 305-318. Zbl1081.05085
5. [5] 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
6. [6] J.A. Telle and A. Proskurowski, Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550, doi: 10.1137/S0895480194275825. Zbl0885.68118

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.