Trees with Equal Domination and Restrained Domination Numbers

  • Authors:

  • Peter Dankelmann;Johannes H. Hattingh;Michael A. Henning;Henda C. Swart

  • Affiliations:

  • School of Mathematical Sciences, University of KwaZulu-Natal, Durban, South Africa 4041;Department of Mathematics and Statistics, Georgia State University, Atlanta, USA 30303-3083;School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, South Africa 3209;School of Mathematical Sciences, University of KwaZulu-Natal, Durban, South Africa 4041

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2006

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Abstract

Let G = (V,E) be a graph and let S V. The set S is a packing in G if the vertices of S are pairwise at distance at least three apart in G. The set S is a dominating set (DS) if every vertex in V 驴 S is adjacent to a vertex in S. Further, if every vertex in V 驴 S is also adjacent to a vertex in V 驴 S, then S is a restrained dominating set (RDS). The domination number of G, denoted by 驴(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by 驴r(G), is the minimum cardinality of a RDS of G. The graph G is 驴-excellent if every vertex of G belongs to some minimum DS of G. A constructive characterization of trees with equal domination and restrained domination numbers is presented. As a consequence of this characterization we show that the following statements are equivalent: (i) T is a tree with 驴(T)=驴r(T); (ii) T is a 驴-excellent tree and T 驴 K2; and (iii) T is a tree that has a unique maximum packing and this set is a dominating set of T. We show that if T is a tree of order n with 驴 leaves, then 驴r(T) 驴 (n + 驴 + 1)/2, and we characterize those trees achieving equality.