In this thesis, we consider the threshold metric dimension problem of graphs, related to and motivated by source detection.
We construct a graph G = (V,E) for a given set of sensors of size m: {s1, s2, ..., sm} and a range k > 0. We want that each node v ∈ V has a unique combination of distances (dk (s1, v),dk (s2, v), ...,dk (sm, v)), where dk is the distance function in a graph limited by the range k (the distance is denoted as being ∞ if the distance is larger than k). Our aim in this thesis is to construct such a graph that is extremal in size, that is: the vertex set V is as large as possible. We shall give such constructions with proof for optimality up to k = 3 and general m
and a different construction with incomplete proof for optimality for general k and m. For any construction we will prove that each vertex is uniquely identified.
Furthermore, we will compare our results to another paper with a similar conclusion about the extremal size of graphs with metric dimension m and a given diameter D.