  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 minimum resolving
set for $G$ is a basis of $G$ and its cardinality is the metric dimension of
$G$. The resolving number of a connected graph $G$ is the minimum $k$, such
that every $k$-set of vertices of $G$ is a resolving set. A connected graph $G$
is called randomly $k$-dimensional if each $k$-set of vertices of $G$ is a
basis. In this paper, along with some properties of randomly $k$-dimensional
graphs, we prove that a connected graph $G$ with at least two vertices is
randomly $k$-dimensional if and only if $G$ is complete graph $K_{k+1}$ or an
odd cycle.
