  A \emph{$2$-partition of a graph $G$} is a function $f:V(G)\rightarrow
\{0,1\}$. A $2$-partition $f$ of a graph $G$ is a \emph{locally-balanced with
an open neighborhood} if for every $v\in V(G)$, $$\left\vert \vert \{u\in
N_{G}(v)\colon\,f(u)=0\}\vert - \vert \{u\in N_{G}(v)\colon\,f(u)=1\}\vert
\right\vert\leq 1.$$ A $2$-partition $f^{\prime}$ of a graph $G$ is a
\emph{locally-balanced with a closed neighborhood} if for every $v\in V(G)$,
$$\left\vert \vert \{u\in N_{G}[v]\colon\,f^{\prime}(u)=0\}\vert - \vert \{u\in
N_{G}[v]\colon\,f^{\prime}(u)=1\}\vert \right\vert\leq 1.$$ In this paper we
prove that the problem of the existence of locally-balanced $2$-partition with
an open (closed) neighborhood is $NP$-complete for some restricted classes of
graphs. In particular, we show that the problem of deciding if a given graph
has a locally-balanced $2$-partition with an open neighborhood is $NP$-complete
for biregular bipartite graphs and even bipartite graphs with maximum degree
$4$, and the problem of deciding if a given graph has a locally-balanced
$2$-partition with a closed neighborhood is $NP$-complete even for subcubic
bipartite graphs and odd graphs with maximum degree $3$. Last results prove a
conjecture of Balikyan and Kamalian.
