  A set $S$ of vertices in a graph is an open packing if (open) neighborhoods
of any two distinct vertices in $S$ are disjoint. In this paper, we consider
the graphs that have a unique maximum open packing. We characterize the trees
with this property by using four local operations such that any nontrivial tree
with a unique maximum open packing can be obtained by a sequence of these
operations starting from $P_2$. We also prove that the decision version of the
open packing number is NP-complete even when restricted to graphs of girth at
least $6$. Finally, we show that the recognition of the graphs with a unique
maximum open packing is polynomially equivalent to the recognition of the
graphs with a unique maximum independent set, and we prove that the complexity
of both problems is not polynomial, unless P=NP.
