Connectivity of $3$-distance graphs

For a simple graph G, the $3$-distance graph, D₃(G), is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $3$ in the graph G. For a connected graph G, we provide some conditions for the connectedness of D₃(G). Also, we characterize all trees and unicyclic graphs with connected $3$-distance graph.