The Metric Dimension of the Wheel-Generated Convex Polytopes Q_n

The family Qn of wheel-generated convex polytopes is obtained from an n-antiprism by adjoining a vertex to each of its two n-gonal faces and joining it to every vertex on the corresponding rim. Theorem 3.1 of H. M. A. Siddiqui and M. Imran, "Computing the metric dimension of wheel related graphs", Appl. Math. Comput. 242 (2014) 624–632 (doi:10.1016/j.amc.2014.06.006) states that dim(Qn) = ⌊(2n+4)/5⌋ for n ≥ 6. The first counterexample is Q11, for which dim(Q11) = 4. The preprint proves that the published expression is exactly the minimum cardinality of a resolving set subject to the restriction that all landmarks lie on one rim, and determines the unrestricted metric dimension for every n ≥ 6: for n ≥ 9, dim(Qn) = ⌈4(n−1)/11⌉, increased by one when n ≡ 9 (mod 11); the cases n = 6, 7, 8 have metric dimensions 3, 3, 4. The lower bound is a cyclic gap argument over the rim landmarks; the upper bound is an explicit construction of period 11. A stability bound for the gap sequences of optimal resolving sets is included. Contents: the preprint (23 pages), a self-contained verification script (breadth-first-search distances; the period-11 construction is checked directly for 11 ≤ n ≤ 500 together with the explicit sets for n = 9, 10; with SciPy installed, exact integer programs prove optimality of both the restricted and the unrestricted values for 6 ≤ n ≤ 20), its full output, a README, and SHA-256 checksums.