Sharp bound on the threshold metric dimension of trees

The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the measuring radius of a sensor is $k$. We give a sharp lower bound on the $\mathrm{Tmd}_k$ of trees, depending only on the number of vertices $n$ and the measuring radius $k$. This sharp lower bound grows linearly in $n$ with leading coefficient $3/(k^2+4k+3+\mathbf{1}\{k\equiv 1\pmod 3\})$, disproving earlier conjectures by Tillquist et al. in arXiv:2106.14314 that suspected $n/(\lfloor k^2/4\rfloor +2k)$ as main order term. We provide a construction for the largest possible trees with a given $\mathrm{Tmd}_k$ value. The proof that our optimal construction cannot be improved relies on edge-rewiring procedures of arbitrary (suboptimal) trees with arbitrary resolving sets, which reveal the structure of how small subsets of sensors measure and resolve certain areas in the tree that we call the attraction of those sensors. The notion of `attraction of sensors' might be useful in other contexts beyond trees to solve related problems. We also provide an improved lower bound on the $\mathrm{Tmd}_k$ of arbitrary trees that takes into account the structural properties of the tree, in particular, the number and length of simple paths of degree-two vertices terminating in leaf vertices. This bound complements arXiv:2106.14314, where only trees without degree-two vertices were considered, except the simple case of a single path.