## TREES WITH EQUAL TOTAL DOMINATION AND TOTAL RESTRAINED DOMINATION NUMBERS

 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

## 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.

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

2000 Mathematics Subject Classification: 05C69.

## References

 [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. [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. [3] F. Harary and M. Livingston, Characterization of tree with equal domination and independent domination numbers, Congr. Numer. 55 (1986) 121-150. [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. [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. [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.