### The Steiner Distance Dimension of Graphs

For a nonempty set $S$ of vertices of a connected graph $G$, the Steiner
distance $d(S)$ of $S$ is the minimum size among all
connected subgraphs whose vertex set contains $S$.
For an ordered set $W=\{w_1, w_2, \cdots, w_k\}$ of vertices
in a connected graph $G$ and a vertex $v$ of $G$,
the Steiner representation $s(v | W)$ of $v$ with respect to
$W$ is the $(2^k-1)$-vector
$$s(v | W) = \left(d_1(v), d_2(v), \cdots, d_k(v), d_{1,2}(v),
d_{1,3}(v),
\cdots, d_{1,2,\cdots, k}(v) \right)$$
where $d_{i_1,i_2, \cdots, i_j}(v)$ is the
Steiner distance $d(\{v, w_{i_1}, w_{i_2}, \cdots, w_{i_j}\})$.
The set $W$ is a Steiner resolving set for $G$
if, for every pair $u,v$ of distinct vertices of $G$, $u$ and $v$ have
distinct representations.
A Steiner resolving set
containing a minimum number of vertices is
called a Steiner basis for $G$. The cardinality of
a Steiner basis is the Steiner (distance) dimension $\dim_S(G)$. In this
paper, we study the Steiner dimension of graphs and determine the Steiner
dimensions of several classes of graphs.