On total domination vertex critical graphs of high connectivity |
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Authors: | Michael A Henning Nader Jafari Rad |
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Institution: | a School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa b Department of Mathematics, Shahrood University of Technology, University Blvd., Shahrood, Iran |
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Abstract: | A graph is called γ-critical if the removal of any vertex from the graph decreases the domination number, while a graph with no isolated vertex is γt-critical if the removal of any vertex that is not adjacent to a vertex of degree 1 decreases the total domination number. A γt-critical graph that has total domination number k, is called k-γt-critical. In this paper, we introduce a class of k-γt-critical graphs of high connectivity for each integer k≥3. In particular, we provide a partial answer to the question “Which graphs are γ-critical and γt-critical or one but not the other?” posed in a recent work W. Goddard, T.W. Haynes, M.A. Henning, L.C. van der Merwe, The diameter of total domination vertex critical graphs, Discrete Math. 286 (2004) 255-261]. |
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Keywords: | Total domination Vertex critical Connectivity Diameter |
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