Acquisition-extremal graphs

A total acquisition move in a weighted graph G moves all weight from a vertex u to a neighboring vertex v, provided that before this move the weight on v is at least the weight on u. The total acquisition number, at(G), is the minimum number of vertices with positive weight that remain in G after a sequence of total acquisition moves, starting with a uniform weighting of the vertices of G. For n ≥ 2, Lampert and Slater showed that at(G) ≤ n+1 3 when G has n vertices, and this is sharp. We characterize the graphs achieving equality: at(G) = |V (G)|+1 3 if and only if G ∈ T ∪ {P2, C5}, where T is the family of trees that can be constructed from P5 by iteratively growing paths with three edges from neighbors of leaves.