Trees with Equal Domination and Restrained Domination Numbers

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 − S is adjacent to a vertex in S. Further, if every vertex in V − S is also adjacent to a vertex in V − S, 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 γr(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 ) = γr(T ); (ii) T is a γ-excellent tree and T 6= 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 ` leaves, then γr(T ) ≤ (n + ` + 1)/2, and we characterize those trees achieving equality.