### 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.