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Total domination and total domination subdivision number of a graph and its complement

Favaron, O ; Sharif University of Technology | 2008

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2007.07.088
  3. Publisher: 2008
  4. Abstract:
  5. A set S of vertices of a graph G = (V, E) with no isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination numberγt (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision numbersdγt (G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We consider graphs of order n ≥ 4, minimum degree δ and maximum degree Δ. We prove that if each component of G and over(G, -) has order at least 3 and G, over(G, -) ≠ C5, then γt (G) + γt (over(G, -)) ≤ frac(2 n, 3) + 2 and if each component of G and over(G, -) has order at least 2 and at least one component of G and over(G, -) has order at least 3, then sdγt (G) + sdγt (over(G, -)) ≤ frac(2 n, 3) + 2. We also give a result on γt (G) + γt (over(G, -)) stronger than a conjecture by Harary and Haynes. © 2007 Elsevier B.V. All rights reserved
  6. Keywords:
  7. Number theory ; Numerical methods ; Problem solving ; Set theory ; Nordhaus-Gaddum inequalities ; Total domination number ; Total domination subdivision number ; Graph theory
  8. Source: Discrete Mathematics ; Volume 308, Issue 17 , 6 September , 2008 , Pages 4018-4023 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0012365X07005833