An antimagic labeling of a graph G is a bijection $f:E(G)\to \{1,\dots ,|E(G)\left|\right\}$ such that the weights $w(x)={\sum }_{y\sim x}f(y)$ distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than $K_2$ admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection $f:V(G)\to \{1,\dots ,|V(G)\left|\right\}$ such that the weight$\omega (x)={\sum }_{y\in N_D(x)}f(y)$ is distinct for each vertex x, where $N_D(x)=\{y\in V(G)|d(x,y)\in D\}$ is the D-neigbourhood set of a vertex x. If $N_D(x)=r$, for every vertex x in G, a graph G is said to be $(D,r)$-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for $D=\left\{1\right\}$, all graphs of order up to 8 concur with the conjecture. We prove that the set of $(D,r)$-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric $(D,r)$-regular that are D-antimagic and examples of disjoint union of non-symmetric non-$(D,r)$-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.