###
The Forcing Dimension of a Graph

For an ordered set $W=\{w_1, w_2, \cdots, w_k\}$ of vertices
and a vertex $v$ in a connected graph $G$,
the (metric) representation of $v$ with respect to $W$
is the $k$-vector
$r(v|W)$ = ($d(v, w_1),$ $ d(v, w_2), $ $ \cdots,$ $ d(v, w_k)$),
where $d(x,y)$ represents the distance between the vertices $x$ and $y$.
The set $W$ is a resolving set for $G$ if distinct vertices of $G$
have distinct representations. A resolving set of minimum cardinality
is a basis for $G$ and the number of vertices in a basis is
its (metric) dimension $\dim (G)$.
For a basis $W$ of $G$, a subset $S$ of $W$
is called a forcing subset of $W$
if $W$ is the unique basis
containing $S$. The forcing
number $f_{G}(W, \dim)$ of $W$ in $G$ is the minimum cardinality of
a forcing subset for $W$, while the forcing dimension
$f(G, \dim)$ of $G$ is
the smallest forcing number among all
bases of $G$. The forcing dimensions of some well-known graphs
are determined. It is shown that for all integers
$a, b$ with $0 \leq a \leq b$ and $b \geq 1$, there exists
a nontrivial connected graph $G$ with $f(G) = a$ and $\dim (G) = b$ if
and only if $\{a, b\} \neq \{0, 1\}$.