  Let $G$ be a graph with nonnegative integer weights. A {\it unit acquisition
move} transfers one unit of weight from a vertex to a neighbor that has at
least as much weight. The {\it unit acquisition number} of a graph $G$, denoted
$a_u(G)$, is the minimum size that the set of vertices with positive weight can
be reduced to via successive unit acquisition moves when starting from the
configuration in which every vertex has weight $1$.
  For a graph $G$ with $n$ vertices and minimum degree $k$, we prove $a_u(G)\le
(n-1)/k$, with equality for complete graphs and $C_5$. Also $a_u(G)$ is at most
the minimum size of a maximal matching in $G$, with equality on an infinite
family of graphs. Furthermore, $a_u(G)$ is bounded by the maximum degree and by
$\sqrt{n-1}$ when $G$ is an $n$-vertex tree with diameter at most $4$. We also
construct arbitrarily large trees with maximum degree $5$ having unit
acquisition number $1$, obtain a linear-time algorithm to compute the
acquisition number of a caterpillar, and show that graphs with diameter $2$
have unit acquisition number $1$ except for $C_5$ and the Petersen graph.
