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The Metric Dimension of Unicyclic Graphs

For an ordered set $W=\{w_1, w_2, \cdots, w_k\}$ of vertices
and a vertex $v$ in a graph $G$,
the 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 containing a minimum number of vertices is called a
basis for $G$ and the number of vertices in a basis is
the (metric) dimension $\dim G$. A connected graph is unicyclic if it
contains
exactly one cycle. For a unicyclic graph $G$, tight bounds
for $\dim G$ are derived. It is shown that all numbers
between these bounds are attainable as the dimension of some
unicyclic graph.