Distance-neighborhood counts in the short-truncation regime

Determine the maximum possible number of vertices within distance at most i, and at exactly distance i, from a vertex in a k-resolving set when k<3i−1, thereby completing the truncated metric-dimension analogues of the distance-neighborhood extremal results.

Background

The paper proves exact formulas for the relevant distance-neighborhood quantities in k-truncated metric dimension when k≥3i−1. The complementary parameter range k<3i−1 is not resolved, and the authors explicitly leave that case for further investigation.