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Discrete Mathematics
Volume 286, Issue 3, 28 September 2004, Pages 255-261
 
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doi:10.1016/j.disc.2004.05.010    How to Cite or Link Using DOI (Opens New Window)
Copyright © 2004 Elsevier B.V. All rights reserved.

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The diameter of total domination vertex critical graphs

Wayne Goddarda, 1, Teresa W. Haynesb, Michael A. Henningc, 2, E-mail The Corresponding Author and Lucas C. van der Merwed

aDepartment of Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa bDepartment of Mathematics, East Tennessee State University, Johnson City, TN 37614-0002, USA cSchool of Mathematics, Statistics, & Information Technology, University of KwaZulu-Natal, Pietermaritzburg 3209, South Africa dDepartment of Mathematics, University of Tennessee in Chattanooga, Chattanooga, TN 37403, USA

Received 23 May 2002; 
revised 25 May 2003; 
accepted 21 May 2004. 
Available online 23 August 2004.

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Abstract

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G-v is less than the total domination number of G. These graphs we call γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We characterize the connected graphs with minimum degree one that are γt-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γt-critical graph for kless-than-or-equals, slant8 and provide an example which shows that the maximum diameter is in general at least 5k/3-O(1).

Keywords: Total domination; Vertex critical; Bounds; Diameter

MSC: 05C69

Article Outline

1. Introduction
2. Graphs with end-vertices
3. Bounds on the diameter
3.1. Constructions
4. Open questions
Acknowledgements
References



Discrete Mathematics
Volume 286, Issue 3, 28 September 2004, Pages 255-261
 
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