  A spanning subgraph of a graph $G$ is called a perfect star packing in $G$ if
every component of the spanning subgraph is isomorphic to the star graph
$K_{1,3}$. An efficient dominating set of graph $G$ is a vertex subset $D$ of
$G$ such that each vertex of $G$ not in $D$ is adjacent to exactly one vertex
from $D$ and any two vertices of $D$ are not adjacent in $G$. Fullerene graph
is a connected plane cubic graph with only pentagonal and hexagonal faces,
which is the molecular graph of carbon fullerene. Clearly, a perfect star
packing in a fullerene graph $G$ on $n$ vertices will exist if and only if $G$
has an efficient dominating set of cardinality $\frac{n}{4}$. The problem of
finding an efficient dominating set is algorithmically hard \cite{Alg_hard}. In
this paper, we give a characterization for a fullerene graph to own a perfect
star packing. And mainly show that it is necessary for a fullerene $G$ owning a
perfect star packing to have order being divisible by $8$. This answers an open
problem asked by Dosli\'{c} et. al. and also shows that a fullerene graph with
an efficient dominating set has $8n$ vertices. By the way, we find some
counterexamples for the necessity of Theorem $14$ in \cite{Doslic} and list
some forbidden configurations to preclude the existence of a perfect star
packing of type $P0$.
