Acquisition-extremal graphs

A total acquisition move   in a weighted graph GG moves all weight from a vertex uu to a neighboring vertex vv, provided that before this move the weight on vv is at least the weight on uu. The total acquisition number  , at(G)at(G), is the minimum number of vertices with positive weight that remain in GG after a sequence of total acquisition moves, starting with a uniform weighting of the vertices of GG. For n≥2n≥2, Lampert and Slater showed that at(G)≤n+13 when GG has nn vertices, and this is sharp. We characterize the graphs achieving equality: at(G)=∣V(G)∣+13 if and only if G∈T∪{P2,C5}G∈T∪{P2,C5}, where TT is the family of trees that can be constructed from P5P5 by iteratively growing paths with three edges from neighbors of leaves.