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On total restrained domination in graphs

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Abstract

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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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). https://doi.org/10.1007/s10587-005-0012-2

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  • DOI: https://doi.org/10.1007/s10587-005-0012-2

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