Characterization of D-antimagic oriented trees

Determine, for every distance set $D\neq\{1\}$, necessary and sufficient conditions under which an oriented tree admits a $D$-antimagic labeling.

Background

For the ordinary distance set D={1}D=\{1\}, the paper proves that an oriented tree is distance antimagic if and only if it is a unidirectional path. The proof uses the existence of sinks and the fact that branching or multiple in-neighbors force repeated neighborhoods or repeated weights.

The paper explicitly states that the corresponding existence problem for other distance sets is largely unknown and formulates the requested necessary-and-sufficient characterization for all D≠{1}D\neq\{1\}.

References

However, the characterizations of the existence of $D$-antimagic trees for other distance sets are still largely unknown. Let $D\neq {1}$ be a distance set. Find necessary and sufficient conditions for a tree to admit a $D$-antimagic labeling.

— D-Antimagic Labelings on Oriented Linear Forests  (2501.05035 - Abrar et al., 9 Jan 2025) in Problem environment near the end of Section 3, Section 3 (Distance Antimagic Labelings on Oriented Trees)