Extend truncated-distance neighborhood extremal results

Investigate and determine the maximum possible number of vertices within distance i of a vertex in a k-truncated metric-dimension graph or k-resolving set when k < 3i − 1.

Background

The paper proves exact formulas for the relevant neighborhood-size extremal quantities when the truncation parameter satisfies k ≥ 3i − 1. The complementary parameter range is not resolved by those arguments, leaving the behavior for k < 3i − 1 as a separate open case.