  An assignment of numbers to the vertices of graph G is closed distinguishing
if for any two adjacent vertices v and u the sum of labels of the vertices in
the closed neighborhood of the vertex v differs from the sum of labels of the
vertices in the closed neighborhood of the vertex u unless they have the same
closed neighborhood (i.e. N[u]=N[v]). The closed distinguishing number of G,
denoted by dis[G], is the smallest integer k such that there is a closed
distinguishing labeling for G using integers from the set[k].Also, for each
vertex $v \in V(G)$, let L(v) denote a list of natural numbers available at v.
A list closed distinguishing labeling is a closed distinguishing labeling f
such that $f(v)\in L(v)$ for each $v \in V(G)$.A graph G is said to be closed
distinguishing k-choosable if every k-list assignment of natural numbers to the
vertices of G permits a list closed distinguishing labeling of G. The closed
distinguishing choice number of G, $dis_{\ell}[G]$, is the minimum number k
such that G is closed distinguishing k-choosable. We show that for each integer
t there is a bipartite graph G such that $dis[G] > t$.It was shown that for
every graph G with $\Delta\geq 2$, $dis[G]\leq dis_{\ell}[G]\leq
\Delta^2-\Delta+1$ and there are infinitely values of $\Delta$ for which G
might be chosen so that $dis[G] =\Delta^2-\Delta+1$. We show that the
difference between $dis[G]$ and $dis_{\ell}[G]$ can be arbitrary large and for
every positive integer t there is a graph G such that $dis_{\ell}[G]-dis[G]\geq
t$. We improve the current upper bound and give some number of upper bounds for
the closed distinguishing choice number by using the Combinatorial
Nullstellensatz. We show that it is $\mathbf{NP}$-complete to decide for a
given planar subcubic graph G, whether dis[G]=2. Also, we prove that for every
$k\geq 3$, it is {\bf NP}-complete to decide whether $dis[G]=k$ for a given
graph G
