Investigate the unresolved truncated-distance regime

Determine the maximum possible numbers of vertices at specified distances from a vertex in a k-resolving set when the truncation parameter satisfies k < 3i − 1, complementing the formulas established for k ≥ 3i − 1.

Background

Section 9 extends the distance-counting results from ordinary metric dimension to k-truncated metric dimension when the truncation threshold is sufficiently large relative to the distance under consideration. The complementary parameter range remains unaddressed.