  A sequence of vertices $(v_1,\, \dots , \,v_k)$ of a graph $G$ is called a
{\it dominating closed neighborhood sequence} if $\{v_1,\, \dots , \,v_k\}$ is
a dominating set of $G$ and $N[v_i]\nsubseteq \cup _{j=1}^{i-1} N[v_j]$ for
every $i$. A graph $G$ is said to be {\it $k-$uniform} if all dominating closed
neighborhood sequences have equal length $k$. Bre{\v s}ar et al. (2014)
characterized $k$-uniform graphs with $k\leq 3$. In this article we extend
their work by giving a complete characterization of all $k$-uniform graphs with
$k\geq 4$.
