Metric Dimension is NP-complete on Graphs of Treewidth Two

Abstract

The metric dimension of a connected graph is the least number of vertices whose distance vectors distinguish all vertices. We show that deciding whether the metric dimension of a graph is at most a given integer is NP-complete on graphs of treewidth at most two. As the problem is polynomial on trees, the graphs of treewidth one, this settles the treewidth threshold of its hardness, which was known to lie between two and twenty-four. The reduction is from satisfiability of formulas in which every literal occurs at most twice. To such a formula with $n$ clauses and $r$ variables it associates a graph $G_φ$ of treewidth at most two with $O((n+r)^2)$ vertices, whose metric dimension is $2(n+r)+5$ if the formula is satisfiable and $2(n+r)+6$ otherwise.