Edge multiset dimension of subdivided complete graphs

An edge of a graph can be identified by its distances to a suitable set of vertices, with each distance measured from the nearer endpoint. If the reports omit the names of these vertices, only a multiset of distances remains, and finding a suitable set becomes more difficult. We determine the asymptotic cost of this loss of information for subdivided complete graphs. Let Kn be the complete graph on n ≥ 3 vertices, and let SL(Kn) replace each edge with a path of L edges, where L ≥ 2 is an integer. For every fixed L ≥ 3, we prove that the edge multiset dimension of SL(Kn) has order n1+1/⌊L/2⌋ as n tends to infinity, with constants depending on L. Retaining the names requires only Θ(n) vertices. Without them, the cost is quadratic at length three, has order n3/2 at lengths four and five, and has smaller exponents at longer fixed lengths. We also settle existence at the two shortest lengths for every n ≥ 3: no selection works at length two, whereas at length three at most n(n−1)/2 vertices suffice. The proofs combine a count of integer vectors with realizable vertex labels. Sidon sets distinguish endpoint pairs at odd lengths, and exact finite certificates complete the length-three construction. Preprint, version 3.0. 49 pages, 17 figures, 4 tables and 2 algorithms, with all appendices included. The exact CSV certificates and reading instructions are available in the accompanying data repository.