2-Domination critical trees upon edge subdivision

A set S of vertices in a graph G is a 2-dominating set of G if every vertex not in S has at least two neighbors in S, where two vertices are neighbors if they are adjacent. The 2-domination number of G, denoted by γ₂ (G), is the minimum cardinality among all 2-dominating sets of G. The graph G is γ₂-q-critical if the smallest subset of edges whose subdivision necessarily increases γ₂ (G) has cardinality q. We characterize the γ₂-2-critical trees.