## ABSTRACT

This chapter considers the notion of acquisition in a graph. A total acquisition move is a transfer of all the weight from a vertex v onto a vertex u, provided that immediately prior to the move, the weight on u is at least the weight on v. The chapter focuses on the behavior of the acquisition number of random graph models, including the binomial random graph and random geometric graphs. It randomizes the process by considering the acquisition number of the path, where vertices have random weights determined by a sequence of independent Poisson variables with unit mean. The chapter discusses the total acquisition number of a path where weights are distributed by a random process. Upper bounds on the acquisition number of trees are shown in terms of the diameter and the order of the graph.