A note on extremal trees for a bound on the double domination number

In a graph G, a vertex is said to dominate itself and its neighbors. A subset D of V(G) is a double dominating set if every vertex of V(G) is dominated at least twice by the vertices of D. The double domination number gamma x2(G) is the minimum cardinality among all double dominating sets of G. Cabrera-Martinez proved that for every nontrivial tree T of order n(T) with & ell;(T) leaves and s(T) support vertices, gamma x2(T) >= 21(n(T) - gamma(T) + & ell;(T) + s(T) + 1), where gamma(T) stands for the domination number of T. In this note, we provide a constructive characterization of trees attaining this bound in response to the problem raised by Cabrera-Martinez. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.