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Trees with Equal Domination and Restrained Domination Numbers

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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 VS is adjacent to a vertex in S. Further, if every vertex in VS is also adjacent to a vertex in VS, then S is a restrained dominating set (RDS). The domination number of G, denoted by γ(G), is the minimum cardinality of a DS of G, while the restrained domination number of G, denoted by γ r (G), is the minimum cardinality of a RDS of G. The graph G is γ-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 γ(T)=γ r (T); (ii) T is a γ-excellent tree and TK2; 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 ℓ leaves, then γ r (T) ≤ (n + ℓ + 1)/2, and we characterize those trees achieving equality.

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Correspondence to Michael A. Henning.

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Dankelmann, P., Hattingh, J.H., Henning, M.A. et al. Trees with Equal Domination and Restrained Domination Numbers. J Glob Optim 34, 597–607 (2006). https://doi.org/10.1007/s10898-005-8565-z

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  • DOI: https://doi.org/10.1007/s10898-005-8565-z

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