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On K-Dimensional Graphs and Their Bases

For an ordered set $W=\{w_1, w_2, \cdots, w_k\}$ of vertices
and a vertex $v$ in a connected graph $G$,
the representation of $v$ with respect to $W$
is the ordered $k$-tuple
$r(v|W)$ = ($d(v, w_1),$ $ 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 every two vertices of $G$
have distinct representations.
A resolving set containing a minimum number of vertices is called a
basis for $G$ and the number of vertices in a basis is its
dimension $\dim(G)$.
If $\dim(G) = k$, then $G$ is said to be $k$-dimensional.
We investigate
how the dimension of a connected graph can be affected by
the addition of a single vertex.
A formula is developed for the dimension of a wheel.
It is shown that for every integer $k \geq2$, there exists a $k$-dimensional
graph with a unique basis and that
for every pair $r$ and $k$ of integers
with $k \geq 2$ and $0 \leq r \leq k$, there
exists a $k$-dimensional graph
$G$ such that there are exactly $r$ vertices that belong to every
basis of $G$.