Sharp bound on the truncated metric dimension of trees

A k-truncated resolving set of a graph is a subset S⊆V of its vertex set such that the vector (d_k(s,v))_s∈S is distinct for each vertex v∈V where d_k(x,y)=min⁡{d(x,y),k+1} is the graph distance truncated at k+1. We think of elements of a k-truncated resolving set as sensors that can measure up to distance k. The k-truncated metric dimension (Tmd_k) of a graph G is the minimum cardinality of a k-truncated resolving set of G. We give a sharp lower bound on Tmd_k for any tree T in terms of its number of vertices |T| and the measuring radius k. Our result is that Tmd_k(T)≥|T|⋅3/(k²+4k+3+1{k≡1(mod3)})+c_k, disproving earlier conjectures by Frongillo et al. that suspected |T|/(⌊k²/4⌋+2k)+c_k^′ as general lower bound, where c_k, c_k^′ are k-dependent constants. We provide a construction for trees with the largest number of vertices with a given 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 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 the result of the above-mentioned work of Frongillo et al., where only trees without degree-two vertices were considered, except the simple case of a single path.