Discussiones Mathematicae Graph Theory 28(1) (2008) 59-66
doi: 10.7151/dmgt.1391

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Xue-Gang Chen

Department of Mathematics
North China Electric Power University
Beijing 102206, China
e-mail: gxc_xdm@163.com

Wai Chee Shiu

Department of Mathematics
Hong Kong Baptist University
224 Waterloo Road, Kowloon Tong, Hong Kong, China

Hong-Yu Chen

The College of Information Science and Engineering
Shandong University of Science and Technology
Qingdao, Shandong Province 266510, China


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.

Keywords: total domination number, total restrained domination number, tree.

2000 Mathematics Subject Classification: 05C69.


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Received 22 September 2006
Revised 24 January 2007
Accepted 24 January 2007