  Let $G$ be a graph, and let $u$, $v$, and $w$ be vertices of $G$. If the
distance between $u$ and $w$ does not equal the distance between $v$ and $w$,
then $w$ is said to resolve $u$ and $v$. The metric dimension of $G$, denoted
$\beta(G)$, is the cardinality of a smallest set $W$ of vertices such that
every pair of vertices of $G$ is resolved by some vertex of $W$. The threshold
dimension of $G$, denoted $\tau(G)$, is the minimum metric dimension among all
graphs $H$ having $G$ as a spanning subgraph. In other words, the threshold
dimension of $G$ is the minimum metric dimension among all graphs obtained from
$G$ by adding edges. If $\beta(G) = \tau(G)$, then $G$ is said to be
irreducible.
  We give two upper bounds for the threshold dimension of a graph, the first in
terms of the diameter, and the second in terms of the chromatic number. As a
consequence, we show that every planar graph of order $n$ has threshold
dimension $O (\log_2 n)$. We show that several infinite families of graphs,
known to have metric dimension $3$, are in fact irreducible. Finally, we show
that for any integers $n$ and $b$ with $1 \leq b < n$, there is an irreducible
graph of order $n$ and metric dimension $b$.
