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Graphs and Combinatorics

, Volume 25, Issue 2, pp 181–196 | Cite as

Total Domination in Partitioned Graphs

  • Allan Frendrup
  • Preben Dahl Vestergaard
  • Anders Yeo
Article
  • 52 Downloads

Abstract

We present results on total domination in a partitioned graph G = (V, E). Let γ t (G) denote the total dominating number of G. For a partition \(V_1, V_2, \ldots , V_k\), k ≥ 2, of V, let γ t (G; V i ) be the cardinality of a smallest subset of V such that every vertex of V i has a neighbour in it and define the following
$$\begin{array}{l} f_t(G; V_1, V_2, \ldots , V_k) = \gamma_t(G) + \gamma_t(G; V_1) + \gamma_t(G; V_2) +\cdots +\gamma_{t}(G;V_k) \\ f_t(G; k) = \max \{ f_{t}(G; V_1, V_2,\ldots , V_k) \mid V_1, V_2, \ldots , V_k {\rm is a partition of } V\} \\ g_t(G; k) = \max\{ \Sigma _{i=1}^{k}\gamma_t(G; V_i) \mid V_1, V_2, \ldots, V_k {\rm is a partition of } V \} \end{array} $$
We summarize known bounds on γ t (G) and for graphs with all degrees at least δ we derive the following bounds for f t (G; k) and g t (G; k).
  1. (i)

    For δ ≥ 2 and k ≥ 3 we prove f t (G; k) ≤ 11|V|/7 and this inequality is best possible.

     
  2. (ii)

    for δ ≥ 3 we prove that f t (G; 2) ≤ (5/4 − 1/372)|V|. That inequality may not be best possible, but we conjecture that f t (G; 2) ≤ 7|V|/6 is.

     
  3. (iii)

    for δ ≥ 3 we prove f t (G; k) ≤  3|V|/2 and this inequality is best possible.

     
  4. (iv)

    for δ ≥ 3 the inequality g t (G; k) ≤ 3|V|/4 holds and is best possible.

     

Keywords

Total domination Partitions and Hypergraphs 

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Copyright information

© Springer Japan 2009

Authors and Affiliations

  • Allan Frendrup
    • 1
  • Preben Dahl Vestergaard
    • 1
  • Anders Yeo
    • 2
  1. 1.Department of Mathematical SciencesAalborg UniversityAalborgDenmark
  2. 2.Department of Computer ScienceUniversity of LondonSurreyUK

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