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On total restrained domination in graphs

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In this paper we initiate the study of total restrained domination in graphs. Let G = (V,E) be a graph. A total restrained dominating set is a set S \( \subseteq \) V where every vertex in V - S is adjacent to a vertex in S as well as to another vertex in V - S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by γ t r (G), is the smallest cardinality of a total restrained dominating set of G. First, some exact values and sharp bounds for γ t r (G) are given in Section 2. Then the Nordhaus-Gaddum-type results for total restrained domination number are established in Section 3. Finally, we show that the decision problem for γ t r (G) is NP-complete even for bipartite and chordal graphs in Section 4.

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  1. G. S. Domke, J. H. Hattingh et al: Restrained domination in graphs. Discrete Math. 203 (1999), 61–69.

    Google Scholar 

  2. M. A. Henning: Graphs with large restrained domination number. Discrete Math. 197/198 (1999), 415–429.

    Google Scholar 

  3. E. A. Nordhaus and J. W. Gaddum: On complementary graphs. Amer. Math. Monthly 63 (1956), 175–177.

    Google Scholar 

  4. F. Jaeger and C. Payan: Relations du type Nordhaus-Gaddum pour le nombre d’absorption d’un granhe simple. C. R. Acad. Sci. Ser. A 274 (1972), 728–730.

    Google Scholar 

  5. M. R. Garey and D. S. Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York, 1979.

    Google Scholar 

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This work was supported by National Natural Sciences Foundation of China (19871036).

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Ma, DX., Chen, XG. & Sun, L. On total restrained domination in graphs. Czech Math J 55, 165–173 (2005).

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