Bounds on the Total Restrained Domination Number of a Graph |
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Authors: | J H Hattingh E Jonck E J Joubert |
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Institution: | 1. Department of Mathematics and Statistics, University Plaza, Georgia State University, Atlanta, GA, 30303, USA 2. Department of Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, 2006, South Africa
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Abstract: | Let G = (V, E) be a graph. A set S í V{S \subseteq V} is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ
tr
(G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ
tr
(G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ
tr
(G) ≤ n − diam(G) − r + 1. |
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