Discussiones Mathematicae Graph Theory 30(3) (2010) 377-383
doi: 10.7151/dmgt.1500

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Joanna Cyman

Department of Technical Physics and Applied Mathematics
Gdańsk University of Technology
Narutowicza 11/12, 80-952 Gdańsk, Poland
e-mail: joana@mif.pg.gda.pl


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 γtc(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 γtc(T) ≥ ⎡[2n/3]⎤. Moreover, we constructively characterize the family of extremal trees T of order n achieving this lower bound.

Keywords: total outer-connected domination number, domination number.

2010 Mathematics Subject Classification: 05C05, 05C69.


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Received 18 March 2009
Revised 27 July 2009
Accepted 17 August 2009