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For a set D of positive integers, we define a vertex set S ⊆ V(G) to be D-independent if u, v ∈ S implies the distance d(u,v) ∉ D. The D-independence number ${\beta }_D\left(G\right)$ is the maximum cardinality of a D-independent set. In particular, the independence number $\beta \left(G\right)={\beta }_1\left(G\right)$. Along with general results we consider, in particular, the odd-independence number ${\beta }_{ODD}\left(G\right)$ where ODD = 1,3,5,....