New bounds on the double domination number of trees

Let G be a nontrivial connected graph with vertex set V(G). A set D subset of V(G) is a double dominating set of G if vertical bar N[nu]boolean AND D vertical bar >= 2 for every vertex nu is an element of V(G), where N[nu] represents the closed neighbourhood of nu. The double domination number of G, denoted by (gamma x2)(G), is the minimum cardinality among all double dominating sets of G. In this note we show that for any nontrivial tree T, n(T) - gamma(T) + l(T) + s(T) + 1/2 (<= gamma x2(T) <= )n(T) + gamma(T) + l(T)/2, where n(T), l(T), s(T) and y(T) represent the order, the number of leaves, the number of support vertices and the classical domination number of T, respectively. In addition, we show that the established upper bound improves a well-known bound and as a consequence, derives two new results. (C) 2022 Elsevier B.V. All rights reserved.