A Note on Completely Disconnecting Trees

Ginsburg and Sands defined a procedure for completely disconnecting graphs: each round, remove at most one edge from each component and at most w edges total. Define fw(g) to be the minimal number of rounds to reduce G to isolated vertices. Prior work of Ginsburg and Sands has determined fw(Pn) for 2 ≤ w ≤ ∞ and the lead-term asymptotics of f∞(Kn) and f2(Kn)). We show that when T is a tree with e edges and max degree at most ∆, f∞(T ) ≤ log ∆ ∆−1 (e− ∆) + ∆. We also give sharp bounds for some specific families of trees.