Bounds on the Total Restrained Domination Number of a Graph

  • Authors:

  • J. H. Hattingh;E. Jonck;E. J. Joubert

  • Affiliations:

  • University Plaza, Georgia State University, Department of Mathematics and Statistics, 30303, Atlanta, GA, USA;University of Johannesburg, Department of Mathematics, P.O. Box 524, 2006, Auckland Park, South Africa;University of Johannesburg, Department of Mathematics, P.O. Box 524, 2006, Auckland Park, South Africa

  • Venue:
  • Graphs and Combinatorics
  • Year:
  • 2010

Quantified Score

Hi-index 0.00

Visualization

Abstract

Let G = (V, E) be a graph. A set $${S \subseteq V}$$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.