On Metric Dimension of Graphs and ‘Their Complements

The metric dimension of a graph \(G\), denoted by \(\text{dim}(G)\), is the minimum number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. For a graph \(G\) and its complement \(\overline{G}\), each of order \(n \geq 4\) and connected, we show that

\[
2 \leq \text{dim}(G) + \text{dim}(\overline{G}) \leq 2(n-3).
\]

It is readily seen that \(\text{dim}(G) + \text{dim}(\overline{G}) = 2\) if and only if \(n = 4\). We characterize graphs satisfying

\[
\text{dim}(G) + \text{dim}(\overline{G}) = 2(n-3)
\]

when \(G\) is a tree or a unicyclic graph.