Truncated-metric-distance extremal counts below the proven threshold

Determine the maximum possible numbers of vertices at prescribed distances from a vertex in a k-resolving set, or within a prescribed distance of such a vertex, in the case k < 3i−1 for k-truncated metric dimension.

Background

The paper proves exact formulas for the relevant distance-neighborhood quantities in k-truncated metric dimension when the truncation parameter satisfies k ≥ 3i−1. The complementary parameter range is not addressed by those results and is explicitly left for further investigation.