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 $\{a, b\} \neq \{0, 1\}$.