The Threshold Dimension of a Graph

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 a graph $G$, 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 \emph{irreducible}; otherwise, we say that $G$ is reducible. If $H$ is a graph having $G$ as a spanning subgraph and such that $\beta(H)=\tau(G)$, then $H$ is called a threshold graph of $G$. The threshold dimension of a graph is expressed in terms of a minimum number of strong products of paths that admits a certain type of embedding of the graph. A sharp upper bound for the threshold dimension of trees is established. It is also shown that the irreducible trees are precisely those of metric dimension at most 2. Moreover, if $T$ is a tree with metric dimension 3 or 4, then $T$ has threshold dimension $2$. It is shown, in these two cases, that a threshold graph for $T$ can be obtained by adding exactly one or two edges to $T$, respectively. However, these results do not extend to trees with metric dimension $5$, i.e., there are trees of metric dimension $5$ with threshold dimension exceeding $2$.