Group distance magic set of group vertex magic graphs

Let \ (G\) be a graph of order \ (n\) and let \ (A\) be an additive Abelian group with identity 0. A mapping \ (l: V (G) A \0\\) is said to be a \ (A\) -vertex magic labeling of \ (G\) if there exists a \ (\) \ (A\) such that \ (w (v) = ₔ ₍₆ (ₕ) l (u) = \) for all \ (v V\) and \ (\) is called a magic constant of \ (\). The group distance magic set of an \ (A\) -vertex magic graph \ (gdms (G, A) \) is defined as \ (gdms (G, A): = \: is a magic constant of some A-vertex magic labeling \\). In this paper, we investigate under what conditions \ (gdms (G, A) \) is a subgroup of \ (A\). We also introduce the concept of the reduced group distance magic set, \ (rgdms (G, A) \), which can be used as a tool to determine \ (gdms (G, A) \).