Magic labelings of distance at most 2

For an arbitrary set of distances $D\subseteq \{0,1,\dots ,d\}$, a graph $G$ is said to be $D$-distance magic if there exists a bijection $f:V\to \{1,2,\dots ,v\}$ and a constant {\sf k} such that for any vertex $x$, ${\sum }_{y\in N_D(x)}f(y)=\mathsf{k}$, where $N_D(x) = {y \in V| d(x,y) \in D}$. In this paper we study some necessary or sufficient conditions for the existence of $D$-distance magic graphs, some of which are generalization of conditions for the existence of $\{1\}$-distance magic graphs. More specifically, we study $D$-distance magic labelings for cycles and $D$-distance magic graphs for $D\subseteq \{0,1,2\}$.