Let G be a graph with vertex set V (G) and edge set E(G). A
set S ⊆ V (G) is a 3-distance independent set of G if dG(v,w) 6= 3
for any two distinct vertices v,w ∈ S. The maximum cardinality of
a 3-distance independent set of G, denoted by alpha^3(G), is called the 3-
distance independence number of G. In this paper, we establish basic
bounds for alpha^3(G), we characterize the graphs for which alpha^3(G) attains
its extreme values, and we introduce the outer 3-distance independent
vertex cover of a graph, obtaining a lower bound for its cardinality in
terms of alpha^3(G).