  For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a
connected graph $G$, the ordered $k$-vector
$r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric)
representation of $v$ with respect to $W$, where $d(x,y)$ is the distance
between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$
if distinct vertices of $G$ have distinct representations with respect to $W$.
A resolving set for $G$ with minimum cardinality is called a basis of $G$ and
its cardinality is the metric dimension of $G$.
  A connected graph $G$ is called randomly $k$-dimensional graph if each
$k$-set of vertices of $G$ is a basis of $G$. In this paper, we study randomly
$k$-dimensional graphs and provide some properties of these graphs.
